A307972 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5*A(x)^2.

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 58, 86, 130, 195, 286, 416, 612, 915, 1380, 2076, 3102, 4627, 6932, 10452, 15818, 23931, 36148, 54600, 82642, 125435, 190724, 290116, 441282, 671512, 1023052, 1560780, 2383578, 3642117, 5567202, 8514254, 13031192, 19960712

Offset: 0

Author

Ilya Gutkovskiy, May 08 2019

Keywords

• nonn

Comments

Shifts 5 places left when convolved with itself.

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + 6*x^10 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000108, A007477, A307970, A307971.

Programs

Maple

a:= proc(n) option remember; `if`(n<5, 1,
      add(a(j)*a(n-5-j), j=0..n-5))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2019

Mathematica

terms = 47; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[a[k] a[n - k - 5], {k, 0, n - 5}]; a[0] = a[1] = a[2] = a[3] = a[4] = 1; Table[a[n], {n, 0, 47}]

SageMath

@CachedFunction
def a(n): # a = A307972
    if (n<5): return 1
    else: return sum(a(k)*a(n-k-5) for k in range(n-4))
[a(n) for n in range(51)] # G. C. Greubel, Nov 26 2022

Formula

G.f.: 1/(1 - x/(1 - x^5/(1 - x^5/(1 - x/(1 - x^5/(1 - x^5/(1 - x/(1 - x^5/(1 - x^5/(1 - ...)))))))))), a continued fraction.

Recurrence: a(n+5) = Sum_{k=0..n} a(k)*a(n-k).