8 장 . 재귀 (1)
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8 장 . 재귀 (1). 이산수학 및 응용 하병현 [email protected]. 목차. 8.1 재귀적으로 정의된 수열 8.2 반복을 사용한 재귀식 풀이 8.3 상수 계수를 갖는 2 계 선형동차 재귀식 8.4 일반적인 재귀적 정의. 8.1 재귀적으로 정의된 수열. - PowerPoint PPT PresentationTRANSCRIPT
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8. (1) [email protected]
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8.1 8.2 8.3 2 8.4
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8.1 So, Natralists observe, a Flea / Hath smaller Fleas that on him prey, / And these have smaller Fleas to bite em, / And so proceed ad infinitum. Jonathan Swift, 1733
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(Recursion) 3, 5, 7, ?n an (1)n/(n + 1), n b0 1, b1 3bk bk1 + bk2
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a0, a1, a2, (recurrence relation) i k (k i) ak ak1, ak2, , aki . (initial condition) , i a0, a1, ai1 , i k a0, a1, , am (m ).
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ck ck1 + kck2 + 1c0 1, c1 2c2, c3, c4?: ? k 1 , sk 3sk1 1 k 0 , sk+1 3sk 1
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: ?1, 1!, 2!, 3!, , (1)nn!, ( n 0)sk (k)sk1 ( k 1) sn , sn (1)nn! k, k 1 ,sk (1)kk!, sk1 (1)k1(k1)!, (k)sk1 (k)(1)k1(k1)! (1)k(1)k1(k1)! (1)kk! sk
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, ,
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: ?64 C . .
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: 4 ,
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():
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(): mk(ab) k a b ,mk(AC) mk1(AB) + 1 + mk1(BC), , mk k ,mk mk1 + 1 + mk1 2mk1 + 1m1 1
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()64 ?m64 1.844674 10191 ? 5845
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. .1 . ?
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(),
,k k 1 +k 2 11 2 3 4 5 6 1235813
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()Fn n ,Fn Fn1 + Fn2http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
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0 1 , .11 0, 1, 2, 3 ? 0: 1: 0, 1 2: 00, 01, 10, 11 3: 000, 001, 010, 011, 100, 101, 110, 111
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()sn nZnonneg n 11 , s0, s1, s2, s3?, 1, 2, 3, 5., 11 10 ? , s10? ?
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() n 11 i) 0 11 ii) 1 11 i) sn1 ii) 10 sn2 , sn sn1 + sn2.
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{1, 2, 3} 2 ?{1, 2}{3} {1, 3}{2} {2, 3}{1}, n r ?2 Stirling (Stirling number of the second kind)Sn,r
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()S4,1?{1, 2, 3, 4}S4,4?{1}{2}{3}{4}S4,2? i) 2 ii) 1 3 {1, 2}{3, 4} {1, 3}{2, 4} {1, 4}{2, 3}{1}{2, 3, 4} {2}{1, 3, 4} {3}{1, 2, 4} {4}{1, 2, 3}
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()S4,3? 1 , 1 , 2 ,{1}{2}{3, 4}{1}{3}{2, 4}{1}{4}{2, 3}{2}{3}{1, 4}{2}{4}{1, 3}{3}{4}{1, 2}
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() ,Sn,r i) n r ii)
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() {n} {n} S4,3 {4} {1}{4}{2, 3} {2}{4}{1, 3} {3}{4}{1, 2}S3,2 S4,3 {4} {1}{2}{3, 4} {1}{3}{2, 4} {2}{3}{1, 4}3S3,3 , Sn,r Sn1,r1 + rSn1,r