数字串的通项公式

王守恩

  A114752
               
1, 2, 5, 4, 9, 6, 13, 8, 17, 10, 21, 12, 25, 14, 29, 16, 33, 18, 37, 20, 41, 22, 45, 24, 49, 26, 53, 28,
57, 30, 61, 32, 65, 34, 69, 36, 73, 38, 77, 40, 81, 42, 85, 44, 89, 46, 93, 48, 97, 50, 101, 52, 105,
54, 109, 56, 113, 58, 117, 60, 121, 62, 125, 64, 129, 66, 133, 68, 137, 70, 141, 72, 145, 74, .....

\(a(n)=1+n*GCD(2,n)\)

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0, 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, 11, 0, 13, 0, 15, 0, 17, 0, 19, 0, 21, 0, 23, 0, 25, 0, 27, 0, 29, 0, 31, 0, 33, 0, 35, 0,
37, 0, 39, 0, 41, 0, 43, 0, 45, 0, 47, 0, 49, 0, 51, 0, 53, 0, 55, 0, 57, 0, 59, 0, 61, 0, 63, 0, 65, 0, 67, 0, 69, 0,....

\(a(n)=n*Mod(n,2)=n*\sin^2(n\pi/2)=\lfloor\frac{n+1}{2}\rfloor^2-\lfloor\frac{n}{2}\rfloor^2\)

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      A008805        这个也可以。

1,  1,  3,  3,  6,  6, 10, 10, 15, 15, 21, 21, 28, 28, 36, 36, 45, 45, 55, 55, 66, 66, 78, 78, 91, 91, 105, 105,
120, 120, 136, 136, 153, 153, 171, 171, 190, 190, 210, 210, 231, 231, 253, 253, 276, 276, 300, 300, 325,
325, 351, 351, 378, 378, 406, 406, 435, 435, 465, 465, 496, 496, 528, 528, 561, 561, 595, 595, 630, 630,
666, 666, 703, 703, 741, 741, 780, 780, 820, 820, 861, 861, 903, 903, 946, 946, 990, 990,1035,1035,......

\(\D a(n)=\sum_{i=\frac{n}{2}}^n\ \sum_{k=\frac{n}{2}}^i\ 1\)

王守恩

周长为 n,边长为整数的三角形。

{0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21,
19, 24, 21, 27, 24, 30, 27, 33, 30, 37, 33, 40, 37, 44, 40, 48, 44, 52, 48, 56, 52, 61, 56,
65, 61, 70, 65, 75, 70, 80, 75, 85, 80, 91, 85, 96, 91, 102, 96, 108, 102, 114, 108, 120,
114, 127, 120, 133, 127, 140, 133, 147, 140, 154, 147, 161, 154, 169, 161, 176, 169,...

a(n)=[\(\frac{(n + 3 Mod(n, 2))^2}{48}\)]   [ ]表示四舍五入

a(n)=SeriesCoefficient[\(\frac{x^3}{(1 - x^2) (1 - x^3) (1 - x^4)}\) {x, 0, n}]

王守恩

有多少个周长为n,三边长是整数的不等边三角形？

0, 0, 0,  0, 0, 0,  0, 0, 1,  0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16,
14, 19, 16, 21, 19, 24, 21, 27, 24, 30, 27, 33, 30, 37, 33, 40, 37, 44, 40, 48, 44, 52, 48,
56, 52, 61, 56, 65, 61, 70, 65, 75, 70, 80, 75, 85, 80, 91, 85, 96, 91, 102, 96, 108, 102,
114, 108, 120, 114, 127, 120, 133, 127,140, 133, 147, 140, 154,147, 161, 154, 169,....

a(n)=[\(\frac{(n-3GCD(n, 2))^2}{48}\)]   [ ]表示四舍五入

王守恩

注意其中细腻的差别。OEIS 中的通项公式可没有这样简单的。

A347420\(\D\ a(n)=\sum_{j = 0}^n\ b\bigg[ n - j,j\bigg]\)

{1, 2, 5, 14, 45, 164, 667, 2986, 14551, 76498, 430747, 2582448, 16403029, 109918746, 774289169, 5715471606, 44087879137, 354521950932,
2965359744447, 25749723493074, 231719153184019, 2157494726318234, 20753996174222511, 205985762120971168, 2106795754056142537,
22180183920771238674, 240109741178147615325, 2670136853490751411326, 30474937734226392850037, 356673190699101499450180, ......


A347420\(\D\ a(n)=\sum_{j = 0}^n\ b\bigg[ j, n - j\bigg]\)

{1, 2, 5, 14, 45, 164, 667, 2986, 14551, 76498, 430747, 2582448, 16403029, 109918746, 774289169, 5715471606, 44087879137, 354521950932,
2965359744447, 25749723493074, 231719153184019, 2157494726318234, 20753996174222511, 205985762120971168, 2106795754056142537,
22180183920771238674, 240109741178147615325, 2670136853490751411326, 30474937734226392850037, 356673190699101499450180, ......


A005490\(\D\ a(n)=\sum_{j = 0}^{n-1}\ b\bigg[ n - j,j\bigg]\)

{0, 1, 4, 13, 44, 163, 666, 2985, 14550, 76497, 430746, 2582447, 16403028, 109918745, 774289168, 5715471605, 44087879136, 354521950931,
2965359744446, 25749723493073, 231719153184018, 2157494726318233, 20753996174222510, 205985762120971167, 2106795754056142536,
22180183920771238673, 240109741178147615324, 2670136853490751411325, 30474937734226392850036, 356673190699101499450179, ......


A005490\(\D\ a(n)=\sum_{j = 1}^{n}\ b\bigg[j, n - j\bigg]\)

{0, 1, 4, 13, 44, 163, 666, 2985, 14550, 76497, 430746, 2582447, 16403028, 109918745, 774289168, 5715471605, 44087879136, 354521950931,
2965359744446, 25749723493073, 231719153184018, 2157494726318233, 20753996174222510, 205985762120971167, 2106795754056142536,
22180183920771238673, 240109741178147615324, 2670136853490751411325, 30474937734226392850036, 356673190699101499450179, ......


A350589\(\D\ a(n)=\sum_{j = 1}^{n}\ b\bigg[ n - j,j\bigg]\)
,
{0, 1, 3, 9, 30, 112, 464, 2109, 10411, 55351, 314772, 1903878, 12189432, 82274309, 583389847, 4332513061, 33607736990, 271657081128,
2283282938288, 19916981288017, 179994994948647, 1682624910161483, 16247280435775188, 161833756265886822, 1660836884761337248,
17541593588541239321, 190478494654528859051, 2124419805554691421937, 24314398329626458197582, 285333388760241224259008, ......


A350589\(\D\ a(n)=\sum_{j = 0}^{n-1}\ b\bigg[ j, n - j\bigg]\)

{0, 1, 3, 9, 30, 112, 464, 2109, 10411, 55351, 314772, 1903878, 12189432, 82274309, 583389847, 4332513061, 33607736990, 271657081128,
2283282938288, 19916981288017, 179994994948647, 1682624910161483, 16247280435775188, 161833756265886822, 1660836884761337248,
17541593588541239321, 190478494654528859051, 2124419805554691421937, 24314398329626458197582, 285333388760241224259008, ......

王守恩

今天是11月23日。

k=2: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229,
k=3: 1, 1, 3, 7, 13, 33, 91, 223, 597, 1753,  4963, 14391, 44413, 137137, 427083, 1382383, 4534981, 14981673, 50719507, 174494983, 605276973, 2135204161,
k=4: 1, 1, 7, 25, 61, 121, 571, 2857, 10585, 30961, 137071, 772441, 3660757, 13925545, 64285411, 382345321, 2121803761, 9777423457, 48698532055, 300809473561,
k=5: 1, 1, 25, 121, 361, 841, 1681, 23185, 186481, 915121, 3338281, 9996361, 105803545, 1098410041, 7394630881, 36583804321, 145787315041, 1366675483105,
k=6: 1, 1, 121, 721,2521, 6721, 15121, 30241, 1869841, 20053441, 119904841, 519158641, 1816574761, 5449167361, 232476185761, 3522419127361, 29720014918561,
......
  
\(\D a(n)=\sum_{j=0}^n\frac{(n - j)!}{(n - k j)!j!\ \ }\)

k=2: OEIS---A000045, 还没有这个通项公式。

k=3: OEIS---A358560, 2022年11月22日创立。

k=4,k=5,k=6,....., OEIS--- 还没有这些数字串。

观察 k=6, 个位数都是 1, 十位数: 5个2,5个4,5个6,5个8,5个0；5个2,5个4,5个6,5个8,5个0；......

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1, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, 675214, 1835421, 4989191, 13562027, 36865412, 100210581, 272400600,
740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245429, 298723530401, 812014744422, 2207284924203, 6000022499693,
16309752131262, 44334502845080, 120513673457548, 327590128640500, 890482293866031, 2420581837980561, 6579823624480555, ......

       这样不也挺好？        Min[HarmonicNumber[k] - n]; Table[a[n], {n, 1, 50}]

王守恩

      A008288       

{1, 01, 01, 001, 001, 0001, 0001, 0001, 00001, 00001, 00001, 00001, 00001, 000001, 000001, 000001, 000001, ...
{1, 03, 05, 007, 009, 0011, 0013, 0015, 00017, 00019, 00021, 00023, 00025, 000027, 000029, 000031, 000033, ...
{1, 05, 13, 025, 041, 0061, 0085, 0113, 00145, 00181, 00221, 00265, 00313, 000365, 000421, 000481, 000545, ...
{1, 07, 25, 063, 129, 0231, 0377, 0575, 00833, 01159, 01561, 02047, 02625, 003303, 004089, 004991, 006017, ...
{1, 09, 41, 129, 321, 0681, 1289, 2241, 03649, 05641, 08361, 11969, 16641, 022569, 029961, 039041, 050049, ...
{1, 11, 61, 231, 681, 1683, 3653, 7183, 13073, 22363, 36365, 56695, 85305, 124515, 177045, 246047, 335137, ...

    Table[CoefficientList[Series[(1 + x)^a/(1 - x)^(1 + a), {x, 0, 16}], x], {a, 0, 5}]

王守恩

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393,
196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986,
102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049,...

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