pdf2htmlEX/demo/cheat2.page

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class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Theoretical<span class="_ _0"> </span>Computer<span class="_ _0"> </span>Science<span class="_ _0"> </span>Cheat<span class="_ _0"> </span>Sheet</div><div class="t m0 xca h3 y2 ff2 fs0 fc0 sc0 ls0 ws0">Iden<span class="_ _5"></span>tities Cont.<span class="_ _2e"> </span>T<span class="_ _5"></span>rees</div><div class="t m0 x114 h2 yb ff1 fs0 fc0 sc0 ls0 ws0">38.</div><div class="t m0 x84 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xc7 h3 y3 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 xc7 h3 y6 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 xe2 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x106 h3 yb ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x107 h6 y115 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xf2 h5 y116 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x115 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x116 h3 y3 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x116 h3 y6 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x94 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x10b h3 y3 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x117 h3 y6 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x82 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x10c h3 yb ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x97 h5 y117 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x118 h6 y118 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x118 h5 y116 ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=0</span></div><div class="t m0 x119 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x89 h3 y3 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x89 h3 y6 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x1 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xc3 h3 yb ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x11a h5 y119 ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff8">−</span>k</div><div class="t m0 x7 h3 yb ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">n</span>!</div><div class="t m0 x11b h5 y117 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 xe9 h6 y118 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xe9 h5 y116 ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=0</span></div><div class="t m0 x26 h3 y3 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x25 h3 y6 ff3 fs0 fc0 sc0 ls0 ws0">k<span class="_ _3"></span><span class="ff2">!</span></div><div class="t m0 xea h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x28 h3 y3 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x58 h3 y6 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 xef h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x29 h3 yb ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _18"> </span><span class="ff1">39.</span></div><div class="t m0 xac h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x3d h3 y3 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 xa8 h4 y6 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>n</div><div class="t m0 x14 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x15 h3 yb ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x6c h5 y117 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x16 h6 y118 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x11c h5 y116 ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=0</span></div><div class="t m0 x42 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"><span class="_ _2b"></span></div><div class="t m0 x4c h3 y3 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x30 h3 y6 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x17 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"><span class="_ _2b"></span></div><div class="t m0 x11d h3 y3 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k</div><div class="t m0 xb6 h3 y6 ff2 fs0 fc0 sc0 ls0 ws0">2<span class="ff3">n</span></div><div class="t m0 x18 h6 y114 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x113 h3 yb ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x114 h2 y11a ff1 fs0 fc0 sc0 ls0 ws0">40.</div><div class="t m0 x84 h6 y11b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x8f h3 y11c ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x11e h3 y50 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x86 h6 y11b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xc8 h3 y11a ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x11f h6 y11d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xe3 h5 y11e ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x120 h6 y11f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x109 h3 y120 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 xf3 h3 y50 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x115 h6 y11f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x8b h3 y120 ff3 fs0 fc0 sc0 ls0 ws0">k<span class="_ _7"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 x8b h3 y50 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 x121 h6 y11f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xd6 h4 y121 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff4">−</span>1)</div><div class="t m0 x119 h5 y122 ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff8">−</span>k</div><div class="t m0 x1 h3 y123 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _2f"> </span><span class="ff1">41.</span></div><div class="t m0 x48 h6 y124 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x3b h3 y125 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2c h3 y50 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 xac h6 y124 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x3c h3 y123 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xb5 h6 y126 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xae h5 y11e ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x5d h6 y11f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x3e h3 y120 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 x3e h3 y50 ff3 fs0 fc0 sc0 ls0 ws0">k<span class="_ _7"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 x122 h6 y11f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xa0 h3 y120 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x30 h3 y50 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 xdd h6 y11f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x34 h4 y121 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff4">−</span>1)</div><div class="t m0 x7a h5 y122 ff6 fs1 fc0 sc0 ls0 ws0">m<span class="ff8">−</span>k</div><div class="t m0 x113 h3 y123 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x114 h2 y17 ff1 fs0 fc0 sc0 ls0 ws0">42.</div><div class="t m0 x84 h6 y127 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x11e h3 y14 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>n<span class="_ _8"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 xfc h3 y128 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x123 h6 y127 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xf3 h3 y17 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x6 h5 y129 ff6 fs1 fc0 sc0 ls0 ws0">m</div><div class="t m0 x116 h6 y12a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x116 h5 y12b ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=0</span></div><div class="t m0 x102 h3 y17 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x10b h6 y12c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x124 h3 y14 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k</div><div class="t m0 x10c h3 y12d ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x125 h6 y12c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x10d h3 y17 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _30"> </span><span class="ff1">43.</span></div><div class="t m0 x67 h6 y12c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x126 h3 y14 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>n<span class="_ _8"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 x5f h3 y12d ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x127 h6 y12c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x14 h3 y17 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x2f h5 y129 ff6 fs1 fc0 sc0 ls0 ws0">m</div><div class="t m0 x6b h6 y12a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x6b h5 y12b ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=0</span></div><div class="t m0 xce h3 y17 ff3 fs0 fc0 sc0 ls0 ws0">k<span class="_ _3"></span><span class="ff2">(</span>n<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k<span class="_ _3"></span><span class="ff2">)</span></div><div class="t m0 x43 h6 y12c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x11d h3 y14 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k</div><div class="t m0 x4e h3 y12d ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 xb0 h6 y12c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x113 h3 y17 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x114 h2 y12e ff1 fs0 fc0 sc0 ls0 ws0">44.</div><div class="t m0 x84 h6 y12f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x11e h3 y130 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc7 h3 y131 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 xe1 h6 y12f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xfc h3 y12e ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x93 h6 y132 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x106 h5 y133 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 xe5 h6 y12f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xe4 h3 y130 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 xe4 h3 y131 ff3 fs0 fc0 sc0 ls0 ws0">k<span class="_ _7"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 x8b h6 y12f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xd4 h3 y130 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x95 h3 y131 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x128 h6 y12f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x103 h4 y12e ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff4">−</span>1)</div><div class="t m0 x129 h5 y134 ff6 fs1 fc0 sc0 ls0 ws0">m<span class="ff8">−</span>k</div><div class="t m0 x12a h4 y135 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _31"> </span><span class="ff1">45.<span class="_ _1e"> </span><span class="ff2">(</span></span>n<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>m<span class="ff2">)!</span></div><div class="t m0 x26 h6 y136 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x9 h3 y137 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 xea h3 y138 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x28 h6 y136 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x72 h3 y135 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xb3 h6 y139 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x2b h5 y133 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x2a h6 y12f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x68 h3 y130 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 x68 h3 y131 ff3 fs0 fc0 sc0 ls0 ws0">k<span class="_ _7"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 x12b h6 y12f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xaf h3 y130 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 xb5 h3 y131 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x13 h6 y12f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x12c h4 y12e ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff4">−</span>1)</div><div class="t m0 x40 h5 y134 ff6 fs1 fc0 sc0 ls0 ws0">m<span class="ff8">−</span>k</div><div class="t m0 x32 h4 y135 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _c"> </span><span class="ff2">for </span>n<span class="_ _7"> </span><span class="ff4">≥<span class="_ _7"> </span></span>m<span class="ff2">,</span></div><div class="t m0 x114 h2 y13a ff1 fs0 fc0 sc0 ls0 ws0">46.</div><div class="t m0 x84 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x86 h3 y58 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x11e h4 y13c ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>m</div><div class="t m0 x93 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xf1 h3 y13a ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x12d h6 y13d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x109 h5 y13e ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x116 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x94 h4 y58 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>n</div><div class="t m0 x94 h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k</div><div class="t m0 xd5 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x96 h3 y58 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>n</div><div class="t m0 x118 h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k</div><div class="t m0 xd8 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xc9 h3 y58 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k</div><div class="t m0 xf6 h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x8 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xe8 h3 y13a ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _32"> </span><span class="ff1">47.</span></div><div class="t m0 x12e h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x73 h3 y58 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc h4 y13c ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>m</div><div class="t m0 x5b h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x126 h3 y13a ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x5c h6 y13d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x2d h5 y13e ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 xa8 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x11 h4 y58 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>n</div><div class="t m0 x11 h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k</div><div class="t m0 x15 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x6c h3 y58 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>n</div><div class="t m0 x40 h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k</div><div class="t m0 x111 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x112 h3 y58 ff3 fs0 fc0 sc0 ls0 ws0">m<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>k</div><div class="t m0 xb6 h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x18 h6 y13b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x113 h3 y13a ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x114 h2 y13f ff1 fs0 fc0 sc0 ls0 ws0">48.</div><div class="t m0 x84 h6 y140 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x86 h3 y141 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x11e h3 y142 ff3 fs0 fc0 sc0 ls0 ws0"><span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>m</div><div class="t m0 x12f h6 y140 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x130 h3 y141 ff3 fs0 fc0 sc0 ls0 ws0"><span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>m</div><div class="t m0 x131 h3 y142 ff3 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x116 h6 y140 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x8b h3 y13f ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x10b h6 y143 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x95 h5 y34 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x121 h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x132 h3 y145 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x133 h3 y146 ff3 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x83 h6 y144 ff7 fs0 fc0 sc0 ls8 ws0"></div><div class="t m0 x134 h4 y145 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>k</div><div class="t m0 x4 h3 y146 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x11a h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xf8 h3 y145 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x135 h3 y146 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x136 h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xe8 h3 y147 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _33"> </span><span class="ff1">49.</span></div><div class="t m0 x137 h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xaa h3 y145 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x9d h3 y146 ff3 fs0 fc0 sc0 ls0 ws0"><span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>m</div><div class="t m0 x48 h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x2d h3 y145 ff3 fs0 fc0 sc0 ls0 ws0"><span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>m</div><div class="t m0 xa8 h3 y146 ff3 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x138 h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x139 h3 y147 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x3e h6 y148 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x2f h5 y34 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x40 h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x13a h3 y145 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x77 h3 y146 ff3 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x4b h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x111 h4 y145 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>k</div><div class="t m0 x33 h3 y146 ff3 fs0 fc0 sc0 ls0 ws0">m</div><div class="t m0 x13b h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x60 h3 y145 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x61 h3 y146 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x18 h6 y144 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x113 h3 y147 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x52 h3 y3 ff2 fs0 fc0 sc0 ls0 ws0">Ev<span class="_ _5"></span>ery<span class="_ _0"> </span>tree<span class="_ _34"> </span>with<span class="_ _0"> </span><span class="ff3">n</span></div><div class="t m0 x52 h4 y5 ff2 fs0 fc0 sc0 ls0 ws0">v<span class="_ _5"></span>ertices<span class="_ _34"> </span>has<span class="_ _0"> </span><span class="ff3">n<span class="_ _7"> </span><span class="ff4">−<span class="_ _7"> </span></span></span>1</div><div class="t m0 x52 h3 y149 ff2 fs0 fc0 sc0 ls0 ws0">edges.</div><div class="t m0 x52 h3 y14a ff2 fs0 fc0 sc0 ls0 ws0">Kraft<span class="_ _35"> </span>inequal-</div><div class="t m0 x52 h3 y14b ff2 fs0 fc0 sc0 ls0 ws0">it<span class="_ _5"></span>y:<span class="_ _36"> </span>If<span class="_ _34"> </span>the<span class="_ _1e"> </span>depths</div><div class="t m0 x52 h3 y14c ff2 fs0 fc0 sc0 ls0 ws0">of<span class="_ _f"> </span>the<span class="_ _17"> </span>lea<span class="_ _5"></span>v<span class="_ _5"></span>es<span class="_ _17"> </span>of</div><div class="t m0 x52 h3 y14d ff2 fs0 fc0 sc0 ls0 ws0">a<span class="_ _11"> </span>binary<span class="_ _11"> </span>tree<span class="_ _11"> </span>are</div><div class="t m0 x52 h3 y14e ff3 fs0 fc0 sc0 ls0 ws0">d</div><div class="t m0 xb1 h5 y14f ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x64 h3 y150 ff3 fs0 fc0 sc0 ls7 ws0">,...,d</div><div class="t m0 x21 h5 y14f ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 xa4 h3 y150 ff2 fs0 fc0 sc0 ls0 ws0">:</div><div class="t m0 xa2 h5 y151 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x54 h6 y152 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x54 h5 y153 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=1</span></div><div class="t m0 xdf h3 y1c ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x20 h5 y154 ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">d</span></div><div class="t m0 xa4 h7 y155 ffa fs2 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa6 h4 y1c ff4 fs0 fc0 sc0 ls0 ws0">≤<span class="_ _7"> </span><span class="ff2">1<span class="ff3">,</span></span></div><div class="t m0 x52 h3 y156 ff2 fs0 fc0 sc0 ls0 ws0">and<span class="_ _7"> </span>equality<span class="_ _7"> </span>holds</div><div class="t m0 x52 h3 y2d ff2 fs0 fc0 sc0 ls0 ws0">only<span class="_ _36"> </span>if<span class="_ _36"> </span>ev<span class="_ _5"></span>ery<span class="_ _36"> </span>in-</div><div class="t m0 x52 h3 y157 ff2 fs0 fc0 sc0 ls0 ws0">ternal<span class="_ _1e"> </span>node<span class="_ _1e"> </span>has<span class="_ _34"> </span>2</div><div class="t m0 x52 h3 y158 ff2 fs0 fc0 sc0 ls0 ws0">sons.</div><div class="t m0 x9b h3 y159 ff2 fs0 fc0 sc0 ls0 ws0">Recurrences</div><div class="t m0 x80 h3 y15a ff2 fs0 fc0 sc0 ls0 ws0">Master metho<span class="_ _3"></span>d:</div><div class="t m0 x3 h4 y3b ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(</span>n<span class="ff2 ls1">)=</span>aT<span class="_ _2"> </span><span class="ff2">(</span>n/b<span class="ff2 ls9">)+</span>f<span class="_ _2"> </span><span class="ff2">(</span>n<span class="ff2">)</span><span class="lsa">,a<span class="_ _a"></span><span class="ff4 ls0">≥<span class="_ _7"> </span><span class="ff2">1<span class="ff3 ls4">,b<span class="_ _2"> </span>&gt;<span class="_ _6"> </span></span>1</span></span></span></div><div class="t m0 x80 h4 y43 ff2 fs0 fc0 sc0 ls0 ws0">If <span class="ff4">∃<span class="ff3 ls1">&gt;</span></span>0 suc<span class="_ _5"></span>h that <span class="ff3">f<span class="_ _2"></span></span>(<span class="ff3">n</span><span class="ls1">)=</span><span class="ff3">O</span>(<span class="ff3">n</span></div><div class="t m0 x96 h5 y65 ff5 fs1 fc0 sc0 ls0 ws0">log</div><div class="t m0 x13c h7 y15b ffa fs2 fc0 sc0 ls0 ws0">b</div><div class="t m0 x10d h5 y65 ff6 fs1 fc0 sc0 ls0 ws0">a<span class="ff8">−</span></div><div class="t m0 x13d h3 y43 ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x80 h3 y15c ff2 fs0 fc0 sc0 ls0 ws0">then</div><div class="t m0 x88 h3 y15d ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(</span>n<span class="ff2 ls1">)=Θ<span class="_ _b"></span>(<span class="_ _b"></span><span class="ff3 ls0">n</span></span></div><div class="t m0 x13e h5 y4a ff5 fs1 fc0 sc0 ls0 ws0">log</div><div class="t m0 x117 h7 y15e ffa fs2 fc0 sc0 ls0 ws0">b</div><div class="t m0 x95 h5 y4a ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 x82 h3 y15f ff2 fs0 fc0 sc0 ls0 ws0">)<span class="ff3">.</span></div><div class="t m0 x80 h3 y160 ff2 fs0 fc0 sc0 ls0 ws0">If <span class="ff3">f<span class="_ _2"></span></span>(<span class="ff3">n</span><span class="ls1">)=Θ<span class="_ _b"></span>(<span class="_ _b"></span><span class="ff3 ls0">n</span></span></div><div class="t m0 xf1 h5 y161 ff5 fs1 fc0 sc0 ls0 ws0">log</div><div class="t m0 x12d h7 y162 ffa fs2 fc0 sc0 ls0 ws0">b</div><div class="t m0 x13f h5 y161 ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 x115 h3 y163 ff2 fs0 fc0 sc0 ls0 ws0">) then</div><div class="t m0 x86 h3 y164 ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(</span>n<span class="ff2 ls1">)=Θ<span class="_ _b"></span>(<span class="_ _b"></span><span class="ff3 ls0">n</span></span></div><div class="t m0 x2 h5 y165 ff5 fs1 fc0 sc0 ls0 ws0">log</div><div class="t m0 x94 h7 y70 ffa fs2 fc0 sc0 ls0 ws0">b</div><div class="t m0 x8b h5 y165 ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 x8c h3 y166 ff2 fs0 fc0 sc0 ls0 ws0">log</div><div class="t m0 x124 h5 y167 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x121 h3 y166 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="ff2">)</span>.</div><div class="t m0 x80 h4 y168 ff2 fs0 fc0 sc0 ls0 ws0">If<span class="_ _7"> </span><span class="ff4">∃<span class="ff3 ls1">&gt;</span></span>0<span class="_ _7"> </span>suc<span class="_ _5"></span>h<span class="_ _7"> </span>that<span class="_ _7"> </span><span class="ff3">f<span class="_ _2"></span></span>(<span class="ff3">n</span><span class="ls1">)=Ω<span class="_ _b"></span>(<span class="_ _b"></span><span class="ff3 ls0">n</span></span></div><div class="t m0 x132 h5 y169 ff5 fs1 fc0 sc0 ls0 ws0">log</div><div class="t m0 x125 h7 y16a ffa fs2 fc0 sc0 ls0 ws0">b</div><div class="t m0 x129 h5 y169 ff6 fs1 fc0 sc0 ls0 ws0">a<span class="ff5">+</span></div><div class="t m0 xd8 h3 y16b ff2 fs0 fc0 sc0 ls0 ws0">),</div><div class="t m0 x80 h4 y16c ff2 fs0 fc0 sc0 ls0 ws0">and<span class="_ _0"> </span><span class="ff4">∃<span class="ff3 lsb">c&lt;<span class="_ _5"></span><span class="ff2 ls0">1<span class="_ _0"> </span>suc<span class="_ _5"></span>h<span class="_ _0"> </span>that <span class="ff3">af<span class="_ _2"></span></span>(<span class="ff3">n/b</span>) <span class="ff4">≤<span class="_ _14"> </span><span class="ff3">cf<span class="_ _2"></span></span></span>(<span class="ff3">n</span>)</span></span></span></div><div class="t m0 x80 h3 y16d ff2 fs0 fc0 sc0 ls0 ws0">for large <span class="ff3">n</span>, then</div><div class="t m0 x12f h3 y16e ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(</span>n<span class="ff2 ls1">)=Θ<span class="_ _b"></span>(<span class="_ _b"></span><span class="ff3 ls0">f<span class="_ _2"></span><span class="ff2">(</span>n<span class="ff2">))</span>.</span></span></div><div class="t m0 x80 h3 y16f ff2 fs0 fc0 sc0 ls0 ws0">Substitution<span class="_ _1e"> </span>(example):<span class="_ _e"> </span>Consider<span class="_ _1e"> </span>the</div><div class="t m0 x80 h3 y170 ff2 fs0 fc0 sc0 ls0 ws0">follo<span class="_ _5"></span>wing recurrence</div><div class="t m0 xe1 h3 y171 ff3 fs0 fc0 sc0 ls0 ws0">T</div><div class="t m0 x140 h5 y172 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x100 h3 y173 ff2 fs0 fc0 sc0 ls1 ws0">=2</div><div class="t m0 x123 h5 y174 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x141 h7 y175 ffa fs2 fc0 sc0 ls0 ws0">i</div><div class="t m0 xf3 h4 y173 ff4 fs0 fc0 sc0 ls0 ws0">·<span class="_ _8"> </span><span class="ff3">T</span></div><div class="t m0 x142 h5 y174 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x116 h5 y176 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x94 h3 y173 ff3 fs0 fc0 sc0 ls3 ws0">,T</div><div class="t m0 x95 h5 y172 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xd5 h3 y173 ff2 fs0 fc0 sc0 ls1 ws0">=2<span class="_ _b"></span><span class="ff3 ls0">.</span></div><div class="t m0 x80 h3 y177 ff2 fs0 fc0 sc0 ls0 ws0">Note<span class="_ _0"> </span>that <span class="ff3">T</span></div><div class="t m0 x100 h5 y178 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x143 h3 y179 ff2 fs0 fc0 sc0 lsc ws0">i<span class="_ _3"></span>s<span class="_ _0"> </span>a<span class="_ _3"></span>l<span class="_ _3"></span>way<span class="_ _3"></span>s<span class="_ _0"> </span>a<span class="_ _0"> </span>p<span class="_ _15"></span>owe<span class="_ _3"></span>r<span class="_ _34"> </span>of<span class="_ _34"> </span>two.</div><div class="t m0 x80 h3 y17a ff2 fs0 fc0 sc0 ls0 ws0">Let <span class="ff3">t</span></div><div class="t m0 x11e h5 y17b ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xc5 h3 y17c ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>log</div><div class="t m0 x100 h5 y17d ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xe3 h3 y17c ff3 fs0 fc0 sc0 ls0 ws0">T</div><div class="t m0 x130 h5 y17b ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xe5 h3 y17c ff2 fs0 fc0 sc0 ls0 ws0">.<span class="_ _34"> </span>Then we ha<span class="_ _5"></span>ve</div><div class="t m0 x86 h3 y17e ff3 fs0 fc0 sc0 ls0 ws0">t</div><div class="t m0 x140 h5 y17f ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x100 h3 y180 ff2 fs0 fc0 sc0 ls1 ws0">=2</div><div class="t m0 x123 h5 y181 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x131 h3 y180 ff2 fs0 fc0 sc0 ls2 ws0">+2<span class="_ _9"></span><span class="ff3 ls0">t</span></div><div class="t m0 x6 h5 y17f ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x94 h3 y180 ff3 fs0 fc0 sc0 ls3 ws0">,t</div><div class="t m0 x144 h5 y17f ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x145 h3 y180 ff2 fs0 fc0 sc0 ls1 ws0">=1<span class="_ _b"></span><span class="ff3 ls0">.</span></div><div class="t m0 x80 h3 yba ff2 fs0 fc0 sc0 ls0 ws0">Let<span class="_ _0"> </span><span class="ff3">u</span></div><div class="t m0 x8f h5 y182 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x86 h3 yba ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _0"> </span><span class="ff3">t</span></div><div class="t m0 xe2 h5 y182 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x93 h3 yba ff3 fs0 fc0 sc0 ls0 ws0">/<span class="ff2">2</span></div><div class="t m0 x143 h5 y183 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x130 h3 yba ff2 fs0 fc0 sc0 ls0 ws0">.<span class="_ _12"> </span>Dividing<span class="_ _0"> </span>b<span class="_ _3"></span>oth<span class="_ _0"> </span>sides<span class="_ _0"> </span>of</div><div class="t m0 x80 h3 y184 ff2 fs0 fc0 sc0 ls0 ws0">the previous equation by 2</div><div class="t m0 x82 h5 ybc ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x132 h3 y185 ff2 fs0 fc0 sc0 ls0 ws0">w<span class="_ _5"></span>e get</div><div class="t m0 xe2 h3 y186 ff3 fs0 fc0 sc0 ls0 ws0">t</div><div class="t m0 x93 h5 y187 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x88 h3 y188 ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x146 h5 y189 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x120 h3 y18a ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x2 h3 y18b ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x142 h5 y18c ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x108 h3 y188 ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2 h5 y189 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x10a h3 y18a ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x95 h3 y18b ff3 fs0 fc0 sc0 ls0 ws0">t</div><div class="t m0 x82 h5 y187 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x144 h3 y188 ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x124 h5 y189 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xd5 h3 y18a ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 y18d ff2 fs0 fc0 sc0 ls0 ws0">Substituting w<span class="_ _5"></span>e find</div><div class="t m0 x9a h3 y18e ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 x86 h5 y18f ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x12f h3 y190 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x147 h5 y191 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x147 h5 y192 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x123 h3 y190 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">u</span></div><div class="t m0 x148 h5 y18f ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x149 h3 y190 ff3 fs0 fc0 sc0 lsd ws0">,u</div><div class="t m0 x82 h5 y18f ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x14a h3 y190 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x96 h5 y191 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x96 h5 y192 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x104 h3 y190 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h3 y193 ff2 fs0 fc0 sc0 ls0 ws0">whic<span class="_ _5"></span>h<span class="_ _34"> </span>is<span class="_ _1e"> </span>simply<span class="_ _0"> </span><span class="ff3">u</span></div><div class="t m0 x108 h5 yd3 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x149 h3 y194 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _34"> </span><span class="ff3">i/</span>2.<span class="_ _e"> </span>So<span class="_ _34"> </span>we<span class="_ _0"> </span>find</div><div class="t m0 x80 h3 y195 ff2 fs0 fc0 sc0 ls0 ws0">that<span class="_ _7"> </span><span class="ff3">T</span></div><div class="t m0 x9a h5 yd4 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xc6 h3 y196 ff2 fs0 fc0 sc0 ls0 ws0">has<span class="_ _7"> </span>the closed<span class="_ _7"> </span>form<span class="_ _14"> </span><span class="ff3">T</span></div><div class="t m0 x14b h5 yd4 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xd6 h3 y196 ff2 fs0 fc0 sc0 ls1 ws0">=2</div><div class="t m0 x14c h5 y197 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">2</span></div><div class="t m0 x14d h7 y198 ffa fs2 fc0 sc0 ls0 ws0">i<span class="ffb">−<span class="ffc">1</span></span></div><div class="t m0 x12a h3 y196 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 y199 ff2 fs0 fc0 sc0 ls0 ws0">Summing<span class="_ _34"> </span>factors<span class="_ _34"> </span>(example):<span class="_ _12"> </span>Consider</div><div class="t m0 x80 h3 y19a ff2 fs0 fc0 sc0 ls0 ws0">the follo<span class="_ _5"></span>wing recurrence</div><div class="t m0 x84 h3 y19b ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(</span>n<span class="ff2 ls1">)=3<span class="_ _b"></span><span class="ff3 ls0">T<span class="_ _2"> </span><span class="ff2">(</span>n/<span class="ff2">2)<span class="_ _7"> </span>+<span class="_ _8"> </span></span>n,<span class="_ _c"> </span>T<span class="_ _2"></span><span class="ff2">(1)<span class="_ _7"> </span>=<span class="_ _7"> </span>1</span>.</span></span></div><div class="t m0 x80 h3 y19c ff2 fs0 fc0 sc0 ls0 ws0">Rewrite<span class="_ _1e"> </span>so<span class="_ _0"> </span>that<span class="_ _1e"> </span>all<span class="_ _34"> </span>terms<span class="_ _1e"> </span>in<span class="_ _5"></span>v<span class="_ _5"></span>olving<span class="_ _34"> </span><span class="ff3">T</span></div><div class="t m0 x80 h3 y19d ff2 fs0 fc0 sc0 ls0 ws0">are on the left side</div><div class="t m0 xfc h4 y19e ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(</span>n<span class="ff2">)<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"></span></span>3</span>T<span class="_ _2"> </span><span class="ff2">(</span>n/<span class="ff2">2)<span class="_ _7"> </span>=<span class="_ _7"> </span></span>n.</div><div class="t m0 x80 h3 y19f ff2 fs0 fc0 sc0 ls0 ws0">No<span class="_ _5"></span>w<span class="_ _7"> </span>expand<span class="_ _8"> </span>the<span class="_ _7"> </span>recurrence,<span class="_ _7"> </span>and<span class="_ _8"> </span>choose</div><div class="t m0 x80 h3 y112 ff2 fs0 fc0 sc0 ls0 ws0">a<span class="_ _7"> </span>factor<span class="_ _7"> </span>whic<span class="_ _5"></span>h<span class="_ _7"> </span>mak<span class="_ _5"></span>es<span class="_ _7"> </span>the<span class="_ _7"> </span>left<span class="_ _7"> </span>side<span class="_ _8"> </span>“tele-</div><div class="t m0 x80 h3 y1a0 ff2 fs0 fc0 sc0 ls0 ws0">scop<span class="_ _3"></span>e”</div><div class="t m0 x25 h3 y1a1 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x56 h6 y1a2 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xea h4 y1a1 ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(</span>n<span class="ff2">)<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"></span></span>3</span>T<span class="_ _2"> </span><span class="ff2">(</span>n/<span class="ff2">2)<span class="_ _7"> </span>=<span class="_ _7"> </span></span>n</div><div class="t m0 x10 h6 y1a2 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x25 h3 y1a3 ff2 fs0 fc0 sc0 ls0 ws0">3</div><div class="t m0 x56 h6 y41 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xea h4 y1a3 ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(</span>n/<span class="ff2">2)<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"></span></span>3</span>T<span class="_ _2"> </span><span class="ff2">(</span>n/<span class="ff2">4)<span class="_ _7"> </span>=<span class="_ _7"> </span></span>n/<span class="ff2">2</span></div><div class="t m0 xcc h6 y41 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x56 h3 y1a4 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x56 h3 y46 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x56 h3 y1a5 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x72 h3 y1a6 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x72 h3 y46 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x72 h3 y1a7 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x14e h3 y1a8 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x14e h3 y46 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x14e h3 y1a9 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x14f h3 y1aa ff2 fs0 fc0 sc0 ls0 ws0">3</div><div class="t m0 x150 h5 y1ab ff5 fs1 fc0 sc0 ls0 ws0">log</div><div class="t m0 xdb h7 y1ac ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x151 h5 y1ab ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x56 h6 y1ad ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xea h4 y1aa ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(2)<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"></span></span>3</span>T<span class="_ _2"> </span><span class="ff2">(1)<span class="_ _7"> </span>=<span class="_ _7"> </span>2</span></div><div class="t m0 x10e h6 y1ad ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xc4 h3 y1ae ff2 fs0 fc0 sc0 ls0 ws0">Let<span class="_ _1e"> </span><span class="ff3">m<span class="_ _1e"> </span></span>=<span class="_ _1e"> </span>log</div><div class="t m0 x9b h5 y1af ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x45 h3 y1b0 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="ff2">.<span class="_ _17"> </span>Summing<span class="_ _1e"> </span>the<span class="_ _34"> </span>left<span class="_ _1e"> </span>side</span></div><div class="t m0 xc4 h4 y1b1 ff2 fs0 fc0 sc0 ls0 ws0">w<span class="_ _5"></span>e<span class="_ _1e"> </span>get<span class="_ _1e"> </span><span class="ff3">T<span class="_ _6"> </span></span>(<span class="ff3">n</span>) <span class="ff4">−<span class="_ _7"> </span></span>3</div><div class="t m0 x46 h5 y79 ff6 fs1 fc0 sc0 ls0 ws0">m</div><div class="t m0 x9d h4 y1b2 ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(1)<span class="_ _11"> </span>=<span class="_ _1e"> </span></span>T<span class="_ _6"> </span><span class="ff2">(</span>n<span class="ff2">) <span class="ff4">−<span class="_ _14"> </span></span>3</span></div><div class="t m0 x152 h5 y79 ff6 fs1 fc0 sc0 ls0 ws0">m</div><div class="t m0 x3f h3 y1b2 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xc4 h4 y1b3 ff3 fs0 fc0 sc0 ls0 ws0">T<span class="_ _6"> </span><span class="ff2">(</span>n<span class="ff2">)<span class="_ _7"> </span><span class="ff4">−<span class="_ _0"> </span></span></span>n</div><div class="t m0 x153 h5 y16a ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x154 h3 y1b4 ff2 fs0 fc0 sc0 ls0 ws0">where<span class="_ _1e"> </span><span class="ff3">k<span class="_ _1e"> </span></span>=<span class="_ _11"> </span>log</div><div class="t m0 x3b h5 ya8 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x5f h4 y1b4 ff2 fs0 fc0 sc0 ls0 ws0">3<span class="_ _1e"> </span><span class="ff4">≈<span class="_ _1e"> </span></span>1<span class="ff3">.</span>58496.</div><div class="t m0 xc4 h3 y1b5 ff2 fs0 fc0 sc0 ls0 ws0">Summing the righ<span class="_ _5"></span>t side we get</div><div class="t m0 xfa h5 y1b6 ff6 fs1 fc0 sc0 ls0 ws0">m<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x153 h6 y1b7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x155 h5 yad ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=0</span></div><div class="t m0 x58 h3 y1b8 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x58 h3 y1b9 ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x9c h5 y1ba ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xef h3 y1bb ff2 fs0 fc0 sc0 ls0 ws0">3</div><div class="t m0 x46 h5 y1bc ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xc1 h3 y1bb ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">n</span></div><div class="t m0 x2a h5 y1b6 ff6 fs1 fc0 sc0 ls0 ws0">m<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x156 h6 y1b7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xe h5 yad ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=0</span></div><div class="t m0 x3b h6 y1bd ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xf h5 y1bc ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 xf h5 y1be ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x74 h6 y1bd ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x3c h5 y1bf ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x11 h3 y1bb ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 xc4 h3 y1c0 ff2 fs0 fc0 sc0 ls0 ws0">Let <span class="ff3">c<span class="_ _7"> </span></span>=</div><div class="t m0 xeb h5 y1c1 ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 xeb h5 y1c2 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x155 h3 y1c3 ff2 fs0 fc0 sc0 ls0 ws0">.<span class="_ _34"> </span>Then we ha<span class="_ _5"></span>ve</div><div class="t m0 x157 h3 y1c4 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x11b h5 y1c5 ff6 fs1 fc0 sc0 ls0 ws0">m<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x151 h6 y1c6 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x158 h5 y1c7 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=0</span></div><div class="t m0 xa h3 y1c8 ff3 fs0 fc0 sc0 ls0 ws0">c</div><div class="t m0 x9 h5 y1c9 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x12e h3 y1c8 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">n</span></div><div class="t m0 xc1 h6 y1ca ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x2b h3 y1cb ff3 fs0 fc0 sc0 ls0 ws0">c</div><div class="t m0 x49 h5 y1cc ff6 fs1 fc0 sc0 ls0 ws0">m</div><div class="t m0 xff h4 y1cb ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1</span></div><div class="t m0 x159 h4 y1cd ff3 fs0 fc0 sc0 ls0 ws0">c<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">1</span></span></div><div class="t m0 xab h6 y1ca ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x12e h3 y1ce ff2 fs0 fc0 sc0 ls1 ws0">=2<span class="_ _b"></span><span class="ff3 ls0">n<span class="ff2">(</span>c</span></div><div class="t m0 xaa h5 y1cf ff5 fs1 fc0 sc0 ls0 ws0">log</div><div class="t m0 xe h7 y1d0 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x48 h5 y1cf ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x5c h4 y1d1 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1)</span></div><div class="t m0 x12e h3 y1d2 ff2 fs0 fc0 sc0 ls1 ws0">=2<span class="_ _b"></span><span class="ff3 ls0">n<span class="ff2">(</span>c</span></div><div class="t m0 xaa h5 y17f ff5 fs1 fc0 sc0 ls0 ws0">(<span class="ff6">k<span class="ff8">−</span></span>1)<span class="_ _6"> </span>log</div><div class="t m0 xf h7 y1d3 ffa fs2 fc0 sc0 ls0 ws0">c</div><div class="t m0 x10e h5 y17f ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x15a h4 y1d4 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1)</span></div><div class="t m0 x12e h3 y1d5 ff2 fs0 fc0 sc0 ls1 ws0">=2<span class="_ _b"></span><span class="ff3 ls0">n</span></div><div class="t m0 xb3 h5 y1d6 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x47 h4 y1d7 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">2<span class="ff3">n,</span></span></div><div class="t m0 xc4 h3 y186 ff2 fs0 fc0 sc0 ls0 ws0">and<span class="_ _0"> </span>so <span class="ff3">T<span class="_ _6"> </span></span>(<span class="ff3">n</span><span class="lse">)=3<span class="_ _20"></span><span class="ff3 ls0">n</span></span></div><div class="t m0 x73 h5 y18c ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 xd h4 y186 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">2<span class="ff3">n</span>.<span class="_ _11"> </span>F<span class="_ _5"></span>ull history re-</span></div><div class="t m0 xc4 h3 y1d8 ff2 fs0 fc0 sc0 ls0 ws0">currences can<span class="_ _7"> </span>often be c<span class="_ _5"></span>hanged to<span class="_ _14"> </span>limited</div><div class="t m0 xc4 h3 y1d9 ff2 fs0 fc0 sc0 ls0 ws0">history ones (example):<span class="_ _1e"> </span>Consider</div><div class="t m0 x151 h3 y1da ff3 fs0 fc0 sc0 ls0 ws0">T</div><div class="t m0 x153 h5 y1db ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x26 h3 y1da ff2 fs0 fc0 sc0 ls1 ws0">=1<span class="_ _5"></span>+</div><div class="t m0 x46 h5 y1dc ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x46 h6 y1dd ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x46 h5 y1de ff6 fs1 fc0 sc0 ls0 ws0">j<span class="_ _3"></span><span class="ff5">=0</span></div><div class="t m0 x47 h3 y1da ff3 fs0 fc0 sc0 ls0 ws0">T</div><div class="t m0 x5b h5 y1db ff6 fs1 fc0 sc0 ls0 ws0">j</div><div class="t m0 x156 h3 y1da ff3 fs0 fc0 sc0 ls3 ws0">,T</div><div class="t m0 x2d h5 y1db ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x74 h3 y1da ff2 fs0 fc0 sc0 ls1 ws0">=1<span class="_ _b"></span><span class="ff3 ls0">.</span></div><div class="t m0 xc4 h3 y1df ff2 fs0 fc0 sc0 ls0 ws0">Note that</div><div class="t m0 xa h3 y1e0 ff3 fs0 fc0 sc0 ls0 ws0">T</div><div class="t m0 x9 h5 ydc ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x72 h3 y1e1 ff2 fs0 fc0 sc0 ls1 ws0">=1<span class="_ _5"></span>+</div><div class="t m0 x15b h5 ye2 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xe h6 y1e2 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xe h5 y1e3 ff6 fs1 fc0 sc0 ls0 ws0">j<span class="_ _3"></span><span class="ff5">=0</span></div><div class="t m0 x3b h3 y1e1 ff3 fs0 fc0 sc0 ls0 ws0">T</div><div class="t m0 x5f h5 ydc ff6 fs1 fc0 sc0 ls0 ws0">j</div><div class="t m0 x15c h3 y1e1 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 xc4 h3 y1e4 ff2 fs0 fc0 sc0 ls0 ws0">Subtracting w<span class="_ _5"></span>e find</div><div class="t m0 x14f h3 yfc ff3 fs0 fc0 sc0 ls0 ws0">T</div><div class="t m0 x15d h5 y1e5 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 xeb h4 yfc ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">T</span></div><div class="t m0 xa h5 y1e5 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x15e h3 yfc ff2 fs0 fc0 sc0 ls1 ws0">=1<span class="_ _5"></span>+</div><div class="t m0 x47 h5 y1e6 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xd h6 y1e7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xd h5 y1e8 ff6 fs1 fc0 sc0 ls0 ws0">j<span class="_ _3"></span><span class="ff5">=0</span></div><div class="t m0 xe h3 yfc ff3 fs0 fc0 sc0 ls0 ws0">T</div><div class="t m0 x15b h5 y1e5 ff6 fs1 fc0 sc0 ls0 ws0">j</div><div class="t m0 x5c h4 yfc ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1<span class="_ _8"> </span></span>−</div><div class="t m0 x3d h5 y1e6 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x3d h6 y1e7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x3d h5 y1e8 ff6 fs1 fc0 sc0 ls0 ws0">j<span class="_ _3"></span><span class="ff5">=0</span></div><div class="t m0 x14 h3 yfc ff3 fs0 fc0 sc0 ls0 ws0">T</div><div class="t m0 xcc h5 y1e5 ff6 fs1 fc0 sc0 ls0 ws0">j</div><div class="t m0 x15e h3 y1e9 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">T</span></div><div class="t m0 x46 h5 y1ea ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x29 h3 y1e9 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 xc4 h3 y1eb ff2 fs0 fc0 sc0 ls0 ws0">And so <span class="ff3">T</span></div><div class="t m0 xfa h5 y1ec ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x9 h3 y1ed ff2 fs0 fc0 sc0 ls1 ws0">=2<span class="_ _b"></span><span class="ff3 ls0">T</span></div><div class="t m0 x29 h5 y1ec ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xb3 h3 y1ed ff2 fs0 fc0 sc0 ls1 ws0">=2</div><div class="t m0 x2a h5 y1ee ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x15f h3 y1ed ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x9f h3 y15a ff2 fs0 fc0 sc0 ls0 ws0">Generating functions:</div><div class="t m0 x160 h3 y3b ff2 fs0 fc0 sc0 ls0 ws0">1.<span class="_ _34"> </span>Multiply<span class="_ _34"> </span>b<span class="_ _3"></span>oth<span class="_ _34"> </span>sides<span class="_ _0"> </span>of<span class="_ _34"> </span>the<span class="_ _34"> </span>equa-</div><div class="t m0 x34 h3 y1ef ff2 fs0 fc0 sc0 ls0 ws0">tion b<span class="_ _5"></span>y <span class="ff3">x</span></div><div class="t m0 x62 h5 y66 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x6f h3 y1f0 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x160 h3 y49 ff2 fs0 fc0 sc0 ls0 ws0">2.<span class="_ _34"> </span>Sum<span class="_ _36"> </span>b<span class="_ _3"></span>oth<span class="_ _36"> </span>sides<span class="_ _36"> </span>o<span class="_ _5"></span>ver<span class="_ _21"> </span>all<span class="_ _36"> </span><span class="ff3">i<span class="_ _36"> </span></span>for</div><div class="t m0 x34 h3 y1f1 ff2 fs0 fc0 sc0 ls0 ws0">whic<span class="_ _5"></span>h the equation is v<span class="_ _5"></span>alid.</div><div class="t m0 x160 h3 y1f2 ff2 fs0 fc0 sc0 ls0 ws0">3.<span class="_ _34"> </span>Cho<span class="_ _15"></span>ose<span class="_ _37"> </span>a<span class="_ _37"> </span>generating<span class="_ _37"> </span>function</div><div class="t m0 x34 h3 y160 ff3 fs0 fc0 sc0 ls0 ws0">G<span class="ff2">(</span>x<span class="ff2">).<span class="_ _34"> </span>Usually </span>G<span class="ff2">(</span>x<span class="ff2 ls1">)=</span></div><div class="t m0 xa4 h6 y1f3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xe0 h5 y1f4 ff8 fs1 fc0 sc0 ls0 ws0">∞</div><div class="t m0 xe0 h5 y1f5 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=0</span></div><div class="t m0 x22 h3 y163 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x24 h5 y161 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x7f h3 y163 ff3 fs0 fc0 sc0 ls0 ws0">g</div><div class="t m0 xa7 h5 y1f6 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x161 h3 y163 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x160 h3 y1f7 ff2 fs0 fc0 sc0 ls0 ws0">3.<span class="_ _34"> </span>Rewrite<span class="_ _0"> </span>the<span class="_ _0"> </span>equation in<span class="_ _0"> </span>terms<span class="_ _0"> </span>of</div><div class="t m0 x34 h3 y1f8 ff2 fs0 fc0 sc0 ls0 ws0">the generating function <span class="ff3">G</span>(<span class="ff3">x</span>).</div><div class="t m0 x160 h3 y1f9 ff2 fs0 fc0 sc0 ls0 ws0">4.<span class="_ _34"> </span>Solve for <span class="ff3">G</span>(<span class="ff3">x</span>).</div><div class="t m0 x160 h3 y1fa ff2 fs0 fc0 sc0 ls0 ws0">5.<span class="_ _34"> </span>The<span class="_ _14"> </span>co<span class="_ _3"></span>efficien<span class="_ _5"></span>t<span class="_ _7"> </span>of<span class="_ _7"> </span><span class="ff3">x</span></div><div class="t m0 x1f h5 y1fb ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x20 h3 y1fc ff2 fs0 fc0 sc0 ls0 ws0">in<span class="_ _7"> </span><span class="ff3">G</span>(<span class="ff3">x</span><span class="lsf">)i<span class="_ _b"></span>s<span class="ff3 ls0">g</span></span></div><div class="t m0 xa7 h5 y1fd ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x161 h3 y1fc ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x9f h3 y1bc ff2 fs0 fc0 sc0 ls0 ws0">Example:</div><div class="t m0 x11d h3 yac ff3 fs0 fc0 sc0 ls0 ws0">g</div><div class="t m0 x13b h5 y88 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x35 h3 yac ff2 fs0 fc0 sc0 ls1 ws0">=2<span class="_ _b"></span><span class="ff3 ls0">g</span></div><div class="t m0 x36 h5 y88 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x63 h3 yac ff2 fs0 fc0 sc0 ls2 ws0">+1<span class="_ _9"></span><span class="ff3 lsa">,g</span></div><div class="t m0 xf0 h5 y88 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x65 h3 yac ff2 fs0 fc0 sc0 ls1 ws0">=0<span class="_ _b"></span><span class="ff3 ls0">.</span></div><div class="t m0 x9f h3 y1fe ff2 fs0 fc0 sc0 ls0 ws0">Multiply and sum:</div><div class="t m0 x162 h6 y1ff ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x163 h5 y200 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">≥<span class="ff5">0</span></span></div><div class="t m0 xec h3 y201 ff3 fs0 fc0 sc0 ls0 ws0">g</div><div class="t m0 xb6 h5 y202 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x35 h3 y201 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 xb0 h5 y203 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x164 h3 y201 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x19 h6 y204 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x19 h5 y200 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">≥<span class="ff5">0</span></span></div><div class="t m0 x64 h3 y201 ff2 fs0 fc0 sc0 ls0 ws0">2<span class="ff3">g</span></div><div class="t m0 xbc h5 y202 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa3 h3 y201 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 xdf h5 y203 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x20 h3 y201 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x66 h6 y204 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x66 h5 y200 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">≥<span class="ff5">0</span></span></div><div class="t m0 xa5 h3 y201 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 xc0 h5 y203 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x22 h3 y201 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x9f h3 y205 ff2 fs0 fc0 sc0 ls0 ws0">W<span class="_ _5"></span>e<span class="_ _8"> </span>c<span class="_ _5"></span>h<span class="_ _5"></span>o<span class="_ _3"></span>ose<span class="_ _8"> </span><span class="ff3">G</span>(<span class="ff3">x</span><span class="ls1">)=</span></div><div class="t m0 xbb h6 y1c7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x54 h5 y206 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">≥<span class="ff5">0</span></span></div><div class="t m0 xdf h3 y207 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 xb2 h5 y208 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x21 h3 y207 ff3 fs0 fc0 sc0 ls0 ws0">g</div><div class="t m0 xa4 h5 y209 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xd1 h3 y207 ff2 fs0 fc0 sc0 ls0 ws0">.<span class="_ _34"> </span>Rewrite</div><div class="t m0 x9f h3 y20a ff2 fs0 fc0 sc0 ls0 ws0">in terms of <span class="ff3">G</span>(<span class="ff3">x</span>):</div><div class="t m0 x34 h4 y20b ff3 fs0 fc0 sc0 ls0 ws0">G<span class="ff2">(</span>x<span class="ff2">)<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span></span>g</div><div class="t m0 xb7 h5 y20c ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x7b h3 yba ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x1b h3 y20d ff2 fs0 fc0 sc0 ls1 ws0">=2<span class="_ _b"></span><span class="ff3 ls0">G<span class="ff2">(</span>x<span class="ff2 ls9">)+</span></span></div><div class="t m0 x65 h6 y20e ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x65 h5 y20f ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">≥<span class="ff5">0</span></span></div><div class="t m0 xbf h3 y20d ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 xba h5 y210 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x165 h3 y20d ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x9f h3 yc1 ff2 fs0 fc0 sc0 ls0 ws0">Simplify:</div><div class="t m0 xec h3 y211 ff3 fs0 fc0 sc0 ls0 ws0">G<span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x7b h3 y212 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x6e h3 y213 ff2 fs0 fc0 sc0 ls1 ws0">=2<span class="_ _b"></span><span class="ff3 ls0">G<span class="ff2">(</span>x<span class="ff2 ls9">)+</span></span></div><div class="t m0 x21 h3 y214 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xdf h4 y212 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff3">x</span></span></div><div class="t m0 xe0 h3 y213 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x9f h3 ycd ff2 fs0 fc0 sc0 ls0 ws0">Solv<span class="_ _5"></span>e for <span class="ff3">G</span>(<span class="ff3">x</span>):</div><div class="t m0 x11d h3 y215 ff3 fs0 fc0 sc0 ls0 ws0">G<span class="ff2">(</span>x<span class="ff2 ls1">)=</span></div><div class="t m0 x166 h3 y1db ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x6f h4 yd3 ff2 fs0 fc0 sc0 ls0 ws0">(1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff3">x</span></span>)(1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>2<span class="ff3">x</span>)</div><div class="t m0 x167 h3 y216 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x9f h3 y217 ff2 fs0 fc0 sc0 ls0 ws0">Expand this using partial fractions:</div><div class="t m0 xdd h3 y218 ff3 fs0 fc0 sc0 ls0 ws0">G<span class="ff2">(</span>x<span class="ff2 ls1">)=</span>x</div><div class="t m0 x4f h6 y219 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x63 h3 y21a ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 xb8 h4 y21b ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>2<span class="ff3">x</span></div><div class="t m0 x166 h4 y21c ff4 fs0 fc0 sc0 ls0 ws0">−</div><div class="t m0 x21 h3 y21d ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xdf h4 y21b ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff3">x</span></span></div><div class="t m0 xe0 h6 y21e ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x4e h3 y21f ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">x</span></div><div class="t m0 x4f h6 y220 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x4f h6 y221 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb8 h3 y21f ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x19 h6 y222 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x19 h5 y223 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">≥<span class="ff5">0</span></span></div><div class="t m0 x64 h3 y224 ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 xcf h5 y225 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x166 h3 y224 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x168 h5 y225 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xdf h4 y224 ff4 fs0 fc0 sc0 ls0 ws0">−</div><div class="t m0 x65 h6 y226 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x65 h5 y223 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">≥<span class="ff5">0</span></span></div><div class="t m0 x167 h3 y224 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 xba h5 y225 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x165 h6 y227 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x165 h6 y228 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x4e h3 y229 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x169 h6 y22a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x35 h5 y22b ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff8">≥<span class="ff5">0</span></span></div><div class="t m0 x16a h3 y22c ff2 fs0 fc0 sc0 ls0 ws0">(2</div><div class="t m0 x19 h5 y22d ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x64 h4 y22c ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1)<span class="ff3">x</span></span></div><div class="t m0 x20 h5 y22d ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">+1</span></div><div class="t m0 x23 h3 y22c ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x9f h3 y22e ff2 fs0 fc0 sc0 ls0 ws0">So <span class="ff3">g</span></div><div class="t m0 x16b h5 y22f ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x4d h3 y230 ff2 fs0 fc0 sc0 ls1 ws0">=2</div><div class="t m0 x7b h5 y231 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x60 h4 y230 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1.</span></div></div><div class="pi" data-data='{"ctm":[1.673203,0.000000,0.000000,1.673203,0.000000,0.000000]}'></div></div></div>