数字串的通项公式

王守恩

王守恩 发表于 2020-10-31 17:35
\(\D\sum_{k=1}^{\infty}\frac{k^n}{2^k}\)
{1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, ...

  挺好的数字串！太久了，我来动一动。
LinearRecurrence[{4, -6, 4, -1}, {2, 9, 25, 56}, 21]
{2, 9, 25, 56, 108, 187, 299, 450, 646, 893, 1197, 1564,
2000, 2511, 3103, 3782, 4554, 5425, 6401, 7488, 8692}

王守恩

王守恩 发表于 2020-12-11 11:02
挺好的数字串！太久了，我来动一动。
LinearRecurrence[{4, -6, 4, -1}, {2, 9, 25, 56}, 21]
{2,  ...

A134593
1, 16, 41, 76, 121, 176, 241, 316, 401, 496, 601, 716, 841, 976, 1121, 1276, 1441, 1616,
1801, 1996, 2201, 2416, 2641, 2876, 3121, 3376, 3641, 3916, 4201, 4496, 4801, 5116,
5441, 5776, 6121, 6476, 6841, 7216, 7601, 7996, 8401, 8816, 9241, 9676, 10121, ....

\(\D a(n)=\frac{(1 + \sqrt{n})^5 + (1 - \sqrt{n})^5}{2}\)

注： 蛮好的通项，可惜了，只要把 “5” 改一改，可以出来好多好多数字串(详见《整数三角形》)。

王守恩

王守恩 发表于 2020-12-19 10:46
A134593
1, 16, 41, 76, 121, 176, 241, 316, 401, 496, 601, 716, 841, 976, 1121, 1276, 1441, 1616 ...

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}

\(\D a_{(n+1)}=\lim_{x\to\infty}\frac{\ln\big(\sum_{k=1}^{x}k^n\big)}{\ln(x)}\)

王守恩

王守恩 发表于 2021-1-4 14:05
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26 ...

      A164118      这通项不是挺好？！
    LinearRecurrence[{1, -1, 1, -1}, {0, -1, -2, -1}, n]
0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0,
0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0,
0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0,

王守恩

王守恩 发表于 2021-1-5 16:32
A164118      这通项不是挺好？！
    LinearRecurrence[{1, -1, 1, -1}, {0, -1, -2, -1}, n]
...

A000254\(\D\ \ \ a(n)=\sum_{k=0}^{n-1}\frac{k(n-1)!}{n-k}+n!\)
0, 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576, 10628640, 120543840, 1486442880,
19802759040, 283465647360, 4339163001600, 70734282393600, 1223405590579200,
22376988058521600, 431565146817638400, 8752948036761600000, 186244810780170240000

王守恩

dlpg070 发表于 2020-7-6 14:38
回复王守恩:
由2个基本语句实现2020=?.?的代码
这是一个简单问题,适合做演示,这是初学者与初学者的对话 ...

     谢谢 dlpg070！想你了！

       A001045\(\D\ \ \ \  a(n)=\bigg[\frac{2^n}{3}\bigg]\)       

  0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691,
  87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243,
  89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531, 5726623061,....................

王守恩

王守恩 发表于 2021-4-17 15:08
谢谢 dlpg070！想你了！

       A001045\(\D\ \ \ \  a(n)=\bigg[\frac{2^n}{3}\bigg]\)       

将 n 个不同整数排成一排，每两个数之间有一个不等号(< 或 >)，出现 m 个 ＞ 号的排列有几种？

1!=1
2!=1+  1
3!=1+  4  +    1
4!=1+ 11 +   11   +    1
5!=1+ 26 +   66   +   26   +     1
6!=1+ 57 +  302  +  302  +    57    +    1
7!=1+120+ 1191 + 2416 +  1191  +  120  +    1
8!=1+247+ 4293 +15619+ 15619 + 4293 +  247  +  1
9!=1+502+14608+88234+156190+88234+14608+502+1
...................

\(\ \ \ \displaystyle\sum_{k=1}^{m}\frac{(m-k)^{n+m}(n+m+1)!}{\cos(k\pi)k!(n+m+1-k)!}+m^{n+m}\)

{{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{1, 4, 11, 26, 57, 120, 247, 502, 1013, 2036, 4083, 8178, 16369, 32752, 65519,131054,262125,524268,},
{1, 11, 66, 302, 1191, 4293, 14608, 47840, 152637, 478271, 1479726, 4537314, 13824739, 41932745,},
{1, 26,  302, 2416, 15619, 88234, 455192, 2203488, 10187685, 45533450, 198410786, 848090912,},
{1, 57,1191, 15619, 156190, 1310354, 9738114, 66318474,423281535,2571742175,15041229521,},
{1, 120, 4293, 88234, 1310354, 15724248, 162512286, 1505621508, 12843262863, 102776998928,},
{1, 247,14608,455192,9738114,162512286,2275172004,27971176092,311387598411,3207483178157,},
{1, 502, 47840, 2203488, 66318474, 1505621508, 27971176092, 447538817472, 6382798925475,}}

王守恩

王守恩 发表于 2021-4-27 18:29
将 n 个不同整数排成一排，每两个数之间有一个不等号(< 或 >)，出现 m 个 ＞ 号的排列有几种？

1!=1 ...

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

\(\ \ \ k=1,\ \ \ A056542\)
{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,}

..............

王守恩

王守恩 发表于 2021-5-4 09:10
\(\D a_{n}=\lfloor\ (e-\frac{\lfloor\ e*k!\ \rfloor}{k!})*n!\ \rfloor\ \ \ \ a_{1}=0\ \ \ \ a_{n ...

\(\D a(n)=\frac{3*5^n-\cos(n\pi)}{2}\)
{1, 8, 37, 188, 937, 4688, 23437, 117188, 585937, 2929688, 14648437,
73242188, 366210937, 1831054688, 9155273437, 45776367188, .....

王守恩

王守恩 发表于 2021-6-16 09:33
\(\D a(n)=\frac{3*5^n-\cos(n\pi)}{2}\)
{1, 8, 37, 188, 937, 4688, 23437, 117188, 585937, 292968 ...

简单的也没有。LinearRecurrence[{1, 20}, {3, 7}, 23]

{3, 7, 67, 207, 1547, 5687, 36627, 150367, 882907, 3890247, 21548387, 99353327,
530321067, 2517387607, 13123808947, 63471561087, 325947740027, 1595378961767,
8114333762307, 40021912997647, 202308588243787, 1002746848196727, 5048918613072467}