cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A365698 G.f. satisfies A(x) = 1 + x^5 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 22, 31, 47, 76, 126, 207, 331, 517, 801, 1251, 1987, 3206, 5212, 8465, 13677, 21997, 35341, 56937, 92169, 149860, 244274, 398383, 649379, 1058055, 1724575, 2814475, 4600923, 7533150, 12347908, 20252837, 33230545
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A212364, A365699, A365700, A365701, A365702.
Cf. A001006, A023426, A025246, A357308.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k-1, n-5*k)*binomial(n-5*k+1, k)/(n-5*k+1));

Formula

G.f.: A(x) = 2*(1+x^5) / (1+x+sqrt( (1+x)^2 - 4*x*(1+x^5) )).
a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k-1,n-5*k) * binomial(n-5*k+1,k) / (n-5*k+1).

A365696 G.f. satisfies A(x) = 1 + x^4*A(x)^2 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 3, 6, 10, 15, 26, 49, 92, 165, 294, 535, 994, 1852, 3437, 6379, 11905, 22344, 42058, 79260, 149601, 283038, 536806, 1020066, 1941317, 3699922, 7062308, 13500402, 25842489, 49528164, 95031920, 182545222, 351023451, 675678911, 1301838177
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A023427, A215341, A215342, A357308, A365697.
Cf. A112805.

Programs

  • Mathematica
    CoefficientList[Series[(1 + x - Sqrt[1 - 2*x + x^2 - 4*x^4])/(2*x*(1 + x^3)), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 26 2023 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k-1, n-4*k)*binomial(n-2*k+1, k)/(n-2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k) * binomial(n-2*k+1,k) / (n-2*k+1).
From Vaclav Kotesovec, Sep 26 2023: (Start)
G.f.: (1 + x - sqrt(1 - 2*x + x^2 - 4*x^4)) / (2*x*(1 + x^3)).
a(n) ~ 2^(n + 3/2) / (sqrt(Pi) * 3^(3/2) * n^(3/2)). (End)

A365697 G.f. satisfies A(x) = 1 + x^4*A(x)^3 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 4, 8, 13, 19, 38, 79, 153, 273, 509, 999, 1979, 3818, 7331, 14279, 28189, 55599, 109275, 215165, 426093, 846638, 1683215, 3348212, 6673679, 13333171, 26679522, 53437369, 107151335, 215154204, 432586412, 870678377, 1754094266
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A023427, A215341, A215342, A357308, A365696.
Cf. A365245.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k-1, n-4*k)*binomial(n-k+1, k)/(n-k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k) * binomial(n-k+1,k) / (n-k+1).

A357307 a(0) = 1, a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

Original entry on oeis.org

1, 0, 1, 2, 2, 4, 8, 13, 25, 49, 91, 177, 349, 681, 1349, 2693, 5377, 10806, 21820, 44163, 89721, 182868, 373616, 765341, 1571551, 3233690, 6667242, 13772469, 28498419, 59065838, 122606998, 254865837, 530507839, 1105663034, 2307131590, 4819623077, 10079039819, 21099213611, 44211213545
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 23 2022

Keywords

Crossrefs

Cf. A000245, A023426, A023431, A090345, A346503, A346504, A357308.

Programs

  • Mathematica
    a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 38}]
nmax = 38; A[] = 0; Do[A[x] = 1 + x^2 (1 + x A[x]^2)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x^2 * (1 + x * A(x)^2) / (1 - x).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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