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 1-3 of 3 results.

A346073 a(n) = 1 + Sum_{k=0..n-4} a(k) * a(n-k-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 20, 29, 45, 73, 118, 185, 293, 475, 778, 1263, 2047, 3345, 5512, 9085, 14957, 24683, 40918, 67987, 113016, 188053, 313608, 524041, 876657, 1467797, 2460644, 4130893, 6942726, 11678687, 19663068, 33139295, 55904339, 94384167, 159470488
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A000108, A007317, A090344, A216604, A307971, A346074.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 42}]
nmax = 42; A[] = 0; Do[A[x] = 1/(1 - x) + x^4 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Jan 22 2023
  • SageMath
    @CachedFunction
def a(n): # a = A346073
    if (n<4): return 1
    else: return 1 + sum(a(k)*a(n-k-4) for k in range(n-3))
[a(n) for n in range(51)] # G. C. Greubel, Nov 26 2022

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^4 * A(x)^2.
a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(k). - Seiichi Manyama, Jan 22 2023

A360027 a(n) = Sum_{k=0..floor(n/5)} (-1)^k * binomial(n-4*k,k) * Catalan(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, -1, -2, -3, -4, -3, 0, 5, 12, 21, 27, 25, 10, -23, -79, -149, -210, -225, -143, 101, 544, 1153, 1783, 2135, 1714, -81, -3735, -9263, -15724, -20603, -19490, -6485, 24242, 75307, 140955, 200891, 215530, 126527, -132122, -605687
Offset: 0

Views

Author

Seiichi Manyama, Jan 22 2023

Keywords

Links

Crossrefs

Cf. A360024, A360025, A360026.
Cf. A000108, A346074.

Programs

  • PARI
    a(n) = sum(k=0, n\5, (-1)^k*binomial(n-4*k, k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=50, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)^2+4*x^5*(1-x))))

Formula

a(n) = 1 - Sum_{k=0..n-5} a(k) * a(n-k-5).
G.f. A(x) satisfies: A(x) = 1/(1-x) - x^5 * A(x)^2.
G.f.: 2 / ( 1-x + sqrt((1-x)^2 + 4*x^5*(1-x)) ).
D-finite with recurrence (n+5)*a(n) 2*(-n-4)*a(n-1) +(n+3)*a(n-2) +2*(2*n-5)*a(n-5) +4*(-n+3)*a(n-6)=0. - R. J. Mathar, Jan 25 2023

A346077 a(n) = 1 + Sum_{k=1..n-5} a(k) * a(n-k-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 12, 18, 26, 36, 49, 69, 101, 150, 221, 320, 460, 667, 981, 1456, 2161, 3191, 4698, 6932, 10283, 15324, 22870, 34103, 50813, 75770, 113229, 169590, 254340, 381579, 572537, 859511, 1291681, 1943489, 2926980, 4410709, 6649220, 10028570
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A105633, A217282, A307972, A346049, A346074, A346075, A346076.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 5], {k, 1, n - 5}]; Table[a[n], {n, 0, 47}]
nmax = 47; A[] = 0; Do[A[x] = 1/(1 - x) + x^5 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • SageMath
    @CachedFunction
def a(n): # a = A346077
    if (n<6): return 1
    else: return 1 + sum(a(k)*a(n-k-5) for k in range(1,n-4))
[a(n) for n in range(51)] # G. C. Greubel, Nov 27 2022

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^5 * A(x) * (A(x) - 1).
Showing 1-3 of 3 results.

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

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