数列通项公式

northwolves · 发表于 2021-4-5 18:38:28

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

大家帮忙找一个这样的数列

.·.·. · 发表于 2021-4-5 23:38:19

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]

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我是找不到通项公式了……

wayne · 发表于 2021-4-6 16:03:55

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))}

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[讨论] 数列通项公式

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