数字串的通项公式

王守恩

1, 2, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761,
271443,  439204, 710647, 1149851, 1860498, 3010349, 4870847,  7881196, 12752043, 20633239, 33385282, 54018521,  87403803,
141422324, 228826127, 370248451, 599074578, 969323029, 1568397607, 2537720636, 4106118243, 6643838879, 10749957122, ...

\(L_{n}=\bigg[\frac{\sin(\arcsin(i/2) n)}{\cos(\arccos(i/2))}\bigg]\)

王守恩

Treenewbee 网友给出这么一个公式，A245031 可是没有的。谢谢 Treenewbee ！

        \(a(n)=\bigg[\frac{\big(3+(3+\cos(n\pi))(2+\sqrt{6})\big)(5+2\sqrt{6})^n}{96}\bigg]\)

0, 1, 21, 120, 2080, 11781, 203841, 1154440, 19974360, 113123361, 1957283461, 11084934960, 191793804840, 1086210502741,
18793835590881, 106437544333680, 1841604094101520, 10429793134197921, 180458407386358101, 1022013289607062600, ......

王守恩

A007910               

1, 2, 3, 6, 13, 26, 51, 102, 205, 410, 819, 1638, 3277, 6554, 13107, 26214, 52429, 104858, 209715,
419430, 838861, 1677722, 3355443, 6710886, 13421773, 26843546, 53687091, 107374182, ......

\(a_{n}=\bigg[\frac{\cos(\arcsin(3i/4) n)}{\cos(\arcsin(3i/4))}\bigg]\)

王守恩

A002530

0, 1, 1, 3, 4, 11, 15, 41, 56, 153, 209, 571, 780, 2131, 2911, 7953, 10864, 29681, 40545,
110771, 151316, 413403, 564719, 1542841, 2107560, 5757961, 7865521, 21489003, 29354524,
80198051, 109552575, 299303201, 408855776, 1117014753, 1525870529, 4168755811,

\(a_n=\frac{(1+\sqrt{3})^{n-1}}{2^{\lceil n/2\rceil}*\sqrt{3}}\)       
       

王守恩

挺好的一串数！可惜 OEIS 没有！！

1,  1,  2,  2,  4,  6,  10,  15,  25, 40, 65, 104, 169, 273, 442, 714, 1156, 1870, 3026, 4895,
7921, 12816, 20737, 33552, 54289, 87841, 142130, 229970, 372100, 602070, 974170,
1576239, 2550409, 4126648, 6677057, 10803704, 17480761, 28284465, 45765226,.....

\(\D a(n)=\bigg\lceil\sum_{k=0}^n \frac{(1+\sqrt{5})^k}{5*2^k}\bigg\rceil\)

1=1*1,  1=1*2,  2=1*2,  2=1*2,
4=2*2,  6=2*3,  10=2*5,  15=3*5,  
25=5*5, 40=5*8, 65=5*13, 104=8*13,
169=13*13, 273=13*21, 442=13*34, 714=21*34,
1156=34*34, 1870=34*55, 3026=34*89, 4895=55*89,
7921=89*89, 12816=89*144, 20737=89*233, 33552=144*233,
54289=233*233, 87841=233*377, 142130=233*610, 229970=377*610,
372100=610*610, 602070=610*987, 974170=610*1597,1576239=987*1597,
2550409, 4126648, 6677057, 10803704, 17480761, 28284465, 45765226,.....

northwolves

王守恩 发表于 2023-3-15 17:27
挺好的一串数！可惜 OEIS 没有！！

1,  1,  2,  2,  4,  6,  10,  15,  25, 40, 65, 104, 169, 273, 442 ...

$a_n=Round[\frac{1 + (2cos\frac{\pi}{5})^{n+1}}{5}]$

王守恩

   谢谢 northwolves！发挥一下。A006498     没有这 2 个公式。

{1, 1, 2, 4, 6, 9, 15, 25, 40, 64, 104, 169, 273, 441, 714, 1156, 1870, 3025, 4895,
7921, 12816, 20736, 33552, 54289, 87841,  142129, 229970, 372100, 602070,
974169, 1576239, 2550409, 4126648, 6677056, 10803704, 17480761, ......

\(\D a(n)=\bigg\lceil\sum_{k=1}^n \frac{(1+\sqrt{5})^k}{5*2^k}\bigg\rceil\)  或   \(\D a(n)=\bigg[\frac{(2\cos(\pi/5))^n}{5}\bigg]\)

northwolves

王守恩 发表于 2023-3-16 09:26
谢谢 northwolves！发挥一下。A006498     没有这 2 个公式。

{1, 1, 2, 4, 6, 9, 15, 25, 40, 64, 1 ...

王老师要干什么？没看明白

王守恩

1， \(\D a(n)=\bigg[\frac{(2\cos(\pi/5))^n}{5}\bigg]\)

2，\(\D a(n)=\bigg\lceil\sum_{k=1}^n \frac{(1+\sqrt{5})^k}{5*2^k}\bigg\rceil\)     

A006498好像还没有这 2 个公式，特别是(1)，尤为简单。

\(\D\bigg\lceil\sum_{k=1}^n \frac{(2\cos(\pi/5))^k}{5}\bigg\rceil=\bigg[\frac{(2\cos(\pi/5))^{n+2}}{5}\bigg]\)

王守恩

      谢谢 northwolves！     A038597

{0, 13, 28, 49,  104, 147, 181, 189, 224, 351, 361, 388, 392, 507, 549,  588,  676, 756, 832,  1029, 1176, 1323,  1369, 1425,
1448, 1512, 1625, 1792, 1862, 1911, 1922, 2299, 2355, 2521, 2808, 2883, 2888, 3104, 3136, 3185, 3216, 3500, 3721, 3969,
4056, 4103, 4332, 4392, 4459, 4537, 4704, 4887, 5103, 5239, 5291, 5341, 5404, 5408, 5439, 5547, 5733, 6048, 6125, ......

Select[Union@Flatten@Table[Sqrt[b^3 - a^3], {a, 1, 6200}, {b, a, Power[a^3 + 6200^2, (3)^-1]}], IntegerQ[#] &]