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# [讨论] 数列通项公式

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x
$a_{n}=1-\frac{n}{a_{n-1}}$

 23:35:12> a=[2,1] %7 = [2, 1] 23:35:16> a=f(a) %8 = [0, 2] 23:35:18> a=[3,1] %9 = [3, 1] 23:35:22> a=f(a) %10 = [1/3, 2] 23:35:23> a=f(a) %11 = [-8, 3] 23:35:24> a=f(a) %12 = [3/2, 4] 23:35:25> a=f(a) %13 = [-7/3, 5] 23:35:26> a=f(a) %14 = [25/7, 6] 23:35:27> a=f(a) %15 = [-24/25, 7] 23:35:28> a=f(a) %16 = [28/3, 8] 23:35:29> a=f(a) %17 = [1/28, 9] 23:35:29> a=f(a) %18 = [-279, 10] 23:35:30> a=f(a) %19 = [290/279, 11] 23:35:31> a=f(a) %20 = [-1529/145, 12] 23:35:31> a=f(a) %21 = [3414/1529, 13] 23:35:32> a=f(a) %22 = [-8996/1707, 14] 23:35:33> a=f(a) %23 = [34601/8996, 15] 23:35:33> a=f(a) %24 = [-109335/34601, 16] 23:35:34> a=f(a) %25 = [697552/109335, 17] 23:35:34> a=f(a) %26 = [-635239/348776, 18] 23:35:35> a=f(a) %27 = [7261983/635239, 19] 23:35:35> a=f(a) %28 = [-5442797/7261983, 20] 23:35:36> a=f(a) %29 = [157944440/5442797, 21] 23:35:36> a=f(a) %30 = [19101453/78972220, 22] 23:35:36> a=f(a) %31 = [-1797259607/19101453, 23] 23:35:37> a=f(a) %32 = [2255694479/1797259607, 24] 23:35:37> a=f(a) %33 = [-42675795696/2255694479, 25] 23:35:38> 23:35:38> a=f(a) %34 = [50661926075/21337897848, 26] 23:35:39> a=f(a) %35 = [-525461315821/50661926075, 27] 23:35:39> a=f(a) %36 = [1943995245921/525461315821, 28] 23:35:40> a=f(a) %37 = [-13294382912888/1943995245921, 29] 23:35:40> a=f(a) %38 = [35807120145259/6647191456444, 30] 23:35:41> a=f(a) %39 = [-170255815004505/35807120145259, 31] 23:35:41> a=f(a) %40 = [1316083659652793/170255815004505, 32] 23:35:42> a=f(a) %41 = [-4302358235495872/1316083659652793, 33] 23:35:42> a=f(a) %42 = [24524601331845417/2151179117747936, 34] 23:35:42> a=f(a) %43 = [-50766667789332343/24524601331845417, 35] 23:35:43> 23:37:13> a=f(a) %44 = [933652315735767355/50766667789332343, 36] 23:37:14> a=f(a) %45 = [-944714392469529336/933652315735767355, 37] 23:37:14> a=f(a) %46 = [18211751195214344413/472357196234764668, 38] 23:37:14> a=f(a) %47 = [-210179457941477639/18211751195214344413, 39] 23:37:15> a=f(a) %48 = [728680227266515254159/210179457941477639, 40] 23:37:17> a=f(a) %49 = [17562509011973528560/17772688469915006199, 41] 23:37:17> a=f(a) %50 = [-364445203362228365899/8781254505986764280, 42] 23:37:17> a=f(a) %51 = [742039147119659229939/364445203362228365899, 43] 23:37:17> a=f(a) %52 = [-15293549800818388869617/742039147119659229939, 44] 复制代码 我是找不到通项公式了……

 这是有理分式。所以如果设初始值是$a_1=t$,那么 $a_n$一定是关于$t$的有理分式。不妨设$a_{n} = \frac{A_{n}+B_{n}t}{C_{n}+D_{n}t}$,于是得到递推公式$A_{n}=k*A_{n-1}-n*k^2*C_{n-1}, B_{n}=k*B_{n-1}-n*k^2*D_{n-1} ,C_{n}=k*A_{n-1}, D_{n} = k*B_{n-1}$ {1,t} {2,1-2/t} {3,-2-6/(-2+t)} {4,3-6/(1+t)} {5,-(2/3)-10/(3 (-1+t))} {6,10-45/(4+t)} {7,3/10-63/(10 (-1+2 t))} {8,-(77/3)-280/(-11+t)} {9,104/77-3240/(77 (-1+11 t))} {10,-(333/52)-14175/(52 (-38+13 t))} {11,905/333-7700/(37 (13+37 t))} {12,-(3091/905)-299376/(181 (-311+181 t))} {13,14856/3091-8845200/(3091 (269+281 t))} {14,-(14209/7428)-4729725/(1238 (-719+619 t))} {15,125629/14209-392931000/(14209 (2257+1093 t))} {16,-(101715/125629)-1667952000/(17947 (-14297+17947 t))} {17,2237408/101715-6175128960/(6781 (37969+6781 t))} {18,203269/1118704-97692469875/(279676 (-35369+69919 t))} {19,-(21052107/203269)-635262264000/(203269 (-30437+1087 t))} {20,25117487/21052107-29331862560000/(2339123 (-562273+2339123 t))}复制代码

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GMT+8, 2021-4-15 18:09 , Processed in 0.067147 second(s), 16 queries .