数字串的通项公式

王守恩

northwolves 发表于 2025-4-1 15:47
A334286

{1, 2, 10, 465, 190131, 848597563, 43025433375905, 26004966055138634525, 194173310204064149 ...

长得太快了！！！465就把我搞晕了！！

1, {1111,2222,3333,4444},
1, {1111,2222,3334,3444},
1, {1111,2222,3344,3344},
1, {1111,2223,2333,4444},
1, {1111,2223,2334,3444},
1, {1111,2223,2344,3344},
1, {1111,2223,2444,3334},
1, {1111,2233,2233,4444},
1, {1111,2233,2234,3444},
1, {1111,2233,2244,3344},
1, {1111,2233,2334,2444},
1, {1111,2233,2344,2344},
1, {1111,2234,2234,3344},

2, {1112,1222,3333,4444},
2, {1112,1222,3334,3444},
2, {1112,1222,3344,3344},
2, {1112,1223,2333,4444},
2, {1112,1223,2334,3444},
2, {1112,1223,2344,3344},
2, {1112,1223,2444,3334},
2, {1112,1224,2333,3444},
2, {1112,1224,2334,3344},
2, {1112,1224,2344,3334},-
2, {1112,1224,2444,3333},
2, {1112,1233,2233,4444},
2, {1112,1233,2234,3444},
2, {1112,1233,2244,3344},
2, {1112,1233,2334,2444},
2, {1112,1233,2344,2344},
2, {1112,1234,2233,3444},
2, {1112,1234,2234,3344},
2, {1112,1234,2244,3334},
2, {1112,1234,2333,2444},-
2, {1112,1234,2334,2344},
2, {1112,1244,2233,3344},
2, {1112,1244,2234,3334},
2, {1112,1244,2244,3333},
2, {1112,1244,2333,2344},
2, {1112,1244,2334,2334},
2, {1112,1333,2223,4444},
2, {1112,1333,2224,3444},
2, {1112,1333,2234,2444},
2, {1112,1333,2244,2344},-
2, {1112,1334,2223,3444},
2, {1112,1334,2224,3344},
2, {1112,1334,2233,2444},
2, {1112,1334,2234,2344},
2, {1112,1334,2244,2334},
2, {1112,1344,2223,3344},
2, {1112,1344,2224,3334},
2, {1112,1344,2233,2344},
2, {1112,1344,2234,2334},
2, {1112,1344,2244,2333},
2, {1112,1444,2223,3334},-
2, {1112,1444,2224,3333},
2, {1112,1444,2233,2334},
2, {1112,1444,2234,2333},

3, {1113,1222,2333,4444},

4, {1114,1222,2333,3444},

5, {1122,1122,3333,4444},

6, {1123,1123,2233,4444},

7, {1124,1124,2233,3344},

8, {1133,1133,2222,4444},

9, {1134,1134,2222,3344},

10, {1144,1144,2222,3333},
10, {1144,1144,2223,2333},
10, {1144,1144,2233,2233},
10, {1144,1222,1233,3344},
10, {1144,1222,1234,3334},
10, {1144,1222,1244,3333},
10, {1144,1222,1333,2344},
10, {1144,1222,1334,2334},
10, {1144,1222,1344,2333},
10, {1144,1223,1223,3344},
10, {1144,1223,1224,3334},
10, {1144,1223,1233,2344},
10, {1144,1223,1234,2334},
10, {1144,1223,1244,2333},
10, {1144,1223,1333,2244},
10, {1144,1223,1334,2234},
10, {1144,1223,1344,2233},

457, {1222,1233,1334,1444},
458, {1222,1234,1333,1444},
459, {1222,1234,1334,1344},

460, {1223,1223,1334,1444},
461, {1223,1223,1344,1344},

462, {1224,1224,1333,1344},
463, {1224,1224,1334,1334},

464, {1233,1233,1244,1244},

465, {1234,1234,1234,1234},

王守恩

{3, 15, 66, 170, 465, 861, 1652, 2412, 3735, 5995, 8382, 12246, 16289, 20055, 25320, 32776, 42381, 48393, 62890, 74130, 84777, 101453, 118956, 138300, 165425, 191997, 212814, 238322, 260565, 286905, 351664},

Table[n (n + 1) Prime[Prime[n]]/2, {n, 40}]

王守恩

A024810——a(n) = floor( tan(m*Pi/2) ), where m = 1 - 2^(-n).

1, 2, 5, 10, 20, 40, 81, 162, 325, 651, 1303, 2607, 5215, 10430, 20860, 41721, 83443, 166886, 333772, 667544, 1335088, 2670176, 5340353, 10680707, 21361414, 42722829, 85445659, 170891318, 341782637, 683565275, 1367130551,

Table[Floor[Csc[Pi/2^n]], {n, 40}]——这样不也挺好？！

{1, 1, 2, 5, 10, 20, 40, 81, 162, 325, 651, 1303, 2607, 5215, 10430, 20860, 41721, 83443, 166886, 333772, 667544, 1335088, 2670176, 5340353, 10680707, 21361414, 42722829, 85445659, 170891318, 341782637, 683565275, 1367130551,

王守恩

OEIS——A079586——3.359885666243177553172011302918927...

\(\frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{8}+\frac{1}{13}+\frac{1}{21}+\frac{1}{34}+\frac{1}{55}+\frac{1}{89}+\frac{1}{144}+\frac{1}{233}+\frac{1}{377}+\frac{1}{610}+\frac{1}{987}+...... = 3.359885666243177553172011302\)

\(\D\sum_{n=0}^{\infty}\frac{\sqrt{5}}{\cos(n\pi)\phi^{2n+1}-1}\)

3.3598856662431775531720113029189271796889051337319684864955538153251303189966833836154162164567900872970453429288539133041367890171008836795913517330771190785803335503
325077531875998504871797778970060395645092153758927752656733540240331694417992939346109926262579646476518686594497102165589843608814726932495910794738736733785233268774
997627277579468536769185419814676687429987673820969139012177220244052081510942649349513745416672789553444707777758478025963407690748474155579104200675015203410705335.......

王守恩

a(02)。(001+(0)×2)×3=3
a(03)。(003+(1+0)×2)×3=15
a(04)。(007+(4+0)×2)×3=45
a(05)。(013+(10+1+0)×2)×3=105
a(06)。(022+(20+4+0)×2)×3=210
a(07)。(034+(35+10+1+0)×2)×3=378
a(08)。(050+(56+20+4+0)×2)×3=630
a(09)。(070+(084+35+10+1+0)×2)×3=0990
a(10)。(095+(120+56+20+4+0)×2)×3=1485
a(11)。(125+(165+084+35+10+1+0)×2)×3=2145
a(12)。(161+(220+120+56+20+4+0)×2)×3=3003
a(13)。(203+(286+165+084+35+10+1+0)×2)×3=4095
a(14)。(252+(364+220+120+56+20+4+0)×2)×3=5460
a(15)。(308+(455+286+165+084+35+10+1+0)×2)×3=7140

{0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, 2145, 3003, 4095, 5460, 7140, 9180, 11628, 14535, 17955, 21945, 26565, 31878, 37950, 44850, 52650, 61425, 71253, 82215, 94395}

1. Table[(n - 1) (n - 0) (n + 1) (n + 2)/8, {n, 29}]

复制代码

王守恩

Table[8 n^2 - 8 n + 1, {n, 45}]        97——难倒全班的初中几何题！
{1, 17, 49, 97, 161, 241, 337, 449, 577, 721, 881, 1057, 1249, 1457, 1681, 1921, 2177, 2449, 2737, 3041, 3361, 3697, 4049, 4417, 4801, 5201, 5617, 6049, 6497, 6961, 7441, 7937, 8449, 8977, 9521, 10081}

FoldList[#2 + #1 &, 1, 16 Range@45]
{1, 17, 49, 97, 161, 241, 337, 449, 577, 721, 881, 1057, 1249, 1457, 1681, 1921, 2177, 2449, 2737, 3041, 3361, 3697, 4049, 4417, 4801, 5201, 5617, 6049, 6497, 6961, 7441, 7937, 8449, 8977, 9521, 10081}

FoldList[#2 - #1 &, 1, 16 Range@45]
{1, 15, 17, 31, 33, 47, 49, 63, 65, 79, 81, 95, 97, 111, 113, 127, 129, 143, 145, 159, 161, 175, 177, 191, 193, 207, 209, 223, 225, 239, 241, 255, 257, 271, 273, 287, 289, 303, 305, 319, 321, 335, 337, 351, 353, 367}

Sum[(8 n^2 - 8 n + 1)/n!, {n, Infinity}]
-1 + 9 e

Product[(8 n^2 - 8 n)/(8 n^2 - 8 n + 2), {n, 2, Infinity}]
Pi/4

不仅变化无穷！！！还可以让 e 与 Pi 扯上关系！！！

王守恩

northwolves 发表于 2025-3-28 12:34
$a_n=\lfloor \frac{1}{7} \sqrt{\left(38-2 (-1)^n\right) 10^n}\rfloor$

A048375——数字 k, 满足 k^2 可以划分为左右两个非零平方数。不允许使用前导零，允许使用尾随零。
a(1)=7, 7^2=49, 4=2^2, 9=3^2。
a(2)=13, 13^2=169, 16=4^2, 9=3^2。
a(3)=19, 19^2=361, 36=6^2, 1=1^2。
a(4)=35, 35^2=1225, 1=1^2, 225=15^2。
a(5)=38, 38^2=1444, 144=12^2, 4=2^2。
a(6)=41, 41^2=1681, 16=4^2, 81=9^2。
a(7)=57, 57^2=3249, 324=18^2, 9=3^2。
a(8)=65, 65^2=4225, 4=2^2, 225=15^2。
a(9)=70, 70^2=4900, 4=2^2, 900=30^2。
a(10)=125, 125^2=15625, 1=1^2, 5625=75^2。
a(11)=130, 130^2=16900, 16=4^2, 900=30^2。
a(12)=190, 190^2=36100, 36=6^2, 100=10^2。
......
7, 13, 19, 35, 38, 41, 57, 65, 70, 125, 130, 190, 205, 223, 253, 285, 305, 350, 380, 410, 475, 487, 570, 650, 700, 721, 905, 975, 985, 1012, 1201, 1250, 1265, 1300, 1301, 1442, 1518, 1771, 1900, 2024, 2050, 2163, 2225, 2230, 2277, 2402, 2435,

A198035——数字 k, 满足 k^2 可以划分为左右两个非零平方数。不允许使用前导零，不允许使用尾随零。

7, 13, 19, 35, 38, 41, 57, 65, 125, 205, 223, 253, 285, 305, 475, 487, 721, 905, 975, 985, 1012, 1201, 1265, 1301, 1442, 1518, 1771, 2024, 2163, 2225, 2277, 2402, 2435, 3075, 3125, 3925, 4901, 6013, 7045, 7969, 8225, 8855, 9607, 9625, 9805,

求助。这编程公式下载不了？

王守恩

这串数还真没通项公式？！

{1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, 211799312, 997313824, 4809701440, 23758664096, 119952692896, 618884638912, 3257843882624, 17492190577600, 95680443760576}

RecurrenceTable[{a[n + 1] == a[n] + n*a[n - 1], a[0] == a[1] == 1}, a, {n, 25}]