已知数列 {a(n)} 满足 a(1)=0 ，a(n+1)=na(n)+1 ，求数列 {a(n)} 的通项公式

allqe775

已知数列{an}满足：a[1] = 0, a[n+1] = n·a[n] + 1，求数列{an}的通项公式。

王守恩

\(\ \ a_{n}=\lfloor\ (e-1)*n!\ \rfloor\)

\((e-1)=1.718281828...　 \lfloor 　\rfloor表示小数作\ 0。 \)

{1, 3, 10, 41, 206, 1237, 8660, 69281, 623530, 6235301, 68588312,
823059745, 10699776686, 149796873605, 2246953104076, 35951249665217,
611171244308690, 11001082397556421, 209020565553572000, 4180411311071440001}

luyuanhong

王守恩

luyuanhong 发表于 2021-5-2 22:12

\(主帖： a_{1}=0\ \ \ \ a_{n+1}=n*a_{n}+1\)
\(\ \ \ \ a_{n}=\lfloor\ (e-1)*n!\ \rfloor\)
{0, 1, 3, 10, 41, 206, 1237, 8660, 69281, 623530, 6235301, 68588312,
823059745, 10699776686, 149796873605, 2246953104076, 35951249665217,
611171244308690, 11001082397556421, 209020565553572000, 4180411311071440001}
\(扩(1)： a_{1}=0\ \ \ \ a_{n+1}=(n+1)*a_{n}+1\)
\(\ \ \ \ a_{n}=\lfloor\ (e-2)*n!\ \rfloor\)
{0, 1, 4, 17, 86, 517, 3620, 28961, 260650, 2606501, 28671512, 344058145,
4472755886, 62618582405, 939278736076, 15028459777217, 255483816212690,
4598708691828421, 87375465144740000, 1747509302894800001, 36697695360790800022,

\(扩(2)： a_{1}=0\ \ \ \ a_{n+1}=(n+2)*a_{n}+1\)
\(\ \ \ \ a_{n}=\lfloor\ (e-5/2)*n!\ \rfloor\)
{0, 1, 5, 26, 157, 1100, 8801, 79210, 792101, 8713112, 104557345, 1359245486,
19029436805, 285441552076, 4567064833217, 77640102164690, 1397521838964421,
26552914940324000, 531058298806480001, \11152224274936080022, 245348934048593760485,

\(扩(3)： a_{1}=0\ \ \ \ a_{n+1}=(n+3)*a_{n}+1\)
\(\ \ \ \ a_{n}=\lfloor\ (e-8/3)*n!\ \rfloor\)
{0, 1, 6, 37, 260, 2081, 18730, 187301, 2060312, 24723745, 321408686, 4499721605,
67495824076, 1079933185217, 18358864148690, 330459554676421, 6278731538852000,
125574630777040001, 2637067246317840022, 58015479418992480485, 1334356026636827051156,
..............

王守恩

luyuanhong 发表于 2021-5-2 22:12

\(a_{n}=\lfloor\ (e-\frac{\lfloor\ e*k!\ \rfloor}{k!})*n!\ \rfloor\ \ \ \ a_{1}=0\ \ \ \ a_{n+1}=(n+k)*a_{n}+1\ \ \ \ k=1, 2, 3, 4, 5, 6, ...\)

\(\ \ \ k=1,\ \ \ A056542\)
{0, 1, 4, 17, 86, 517, 3620, 28961, 260650, 2606501, 28671512, 344058145,
4472755886, 62618582405, 939278736076, 15028459777217, 255483816212690,
4598708691828421, 87375465144740000, 1747509302894800001, 36697695360790800022,}

\(\ \ \ k=2,\ \ \ A185108\)
{0, 1, 5, 26, 157, 1100, 8801, 79210, 792101, 8713112, 104557345, 1359245486,
19029436805, 285441552076, 4567064833217, 77640102164690, 1397521838964421,
26552914940324000, 531058298806480001, \11152224274936080022, 245348934048593760485,}

\(\ \ \ k=3,\ \ \ A079751\)
{0, 1, 6, 37, 260, 2081, 18730, 187301, 2060312, 24723745, 321408686, 4499721605,
67495824076, 1079933185217, 18358864148690, 330459554676421, 6278731538852000,
125574630777040001, 2637067246317840022, 58015479418992480485, 1334356026636827051156,}

\(\ \ \ k=4,\ \ \ ?\)
{0, 1, 7, 50, 401, 3610, 36101, 397112, 4765345, 61949486, 867292805, 13009392076,
208150273217, 3538554644690, 63693983604421, 1210185688484000, 24203713769680001,
508277989163280022, 11182115761592160485, 257188662516619691156, 6172527900398872587745,}

\(\ \ \ k=5,\ \ \ ?\)
{0, 1, 8, 65, 586, 5861, 64472, 773665, 10057646, 140807045, 2112105676, 33793690817,
574492743890, 10340869390021, 196476518410400, 3929530368208001, 82520137732368022,
1815443030112096485, 41755189692578219156, 1002124552621877259745, 25053113815546931493626,}

\(\ \ \ k=6,\ \ \ ?\)
{0, 1, 9, 82, 821, 9032, 108385, 1409006, 19726085, 295891276, 4734260417, 80482427090,
1448683687621, 27524990064800, 550499801296001, 11560495827216022, 254330908198752485,
5849610888571307156, 140390661325711371745, 3509766533142784293626, 91253929861712391634277,}

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