A307971 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4*A(x)^2.

1, 1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 20, 29, 44, 70, 112, 175, 272, 430, 690, 1107, 1766, 2822, 4542, 7347, 11886, 19222, 31150, 50647, 82518, 134542, 219542, 358808, 587430, 962898, 1579686, 2593967, 4264292, 7017800, 11559548, 19055420, 31437318, 51908076, 85775954, 141841207

Offset

0,6

Comments

Shifts 4 places left when convolved with itself.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..4000

Formula

G.f.: 1/(1 - x/(1 - x^4/(1 - x^4/(1 - x/(1 - x^4/(1 - x^4/(1 - x/(1 - x^4/(1 - x^4/(1 - ...)))))))))), a continued fraction.

Recurrence: a(n+4) = Sum_{k=0..n} a(k)*a(n-k).

G.f.: (1 - sqrt(1 - 4*x^4 - 4*x^5 - 4*x^6 - 4*x^7))/(2*x^4). - Vaclav Kotesovec, Sep 27 2023

Example

G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 8*x^9 + 13*x^10 + ...

Maple

a:= proc(n) option remember; `if`(n<4, 1,
      add(a(j)*a(n-4-j), j=0..n-4))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2019

Mathematica

terms = 44; A[_] = 0; Do[A[x_] = 1 + x + x^2 + x^3 + x^4 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; a[0] = a[1] = a[2] = a[3] = 1; Table[a[n], {n, 0, 44}]
CoefficientList[Series[(1 - Sqrt[1 - 4*x^4 - 4*x^5 - 4*x^6 - 4*x^7])/(2*x^4), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 27 2023 *)

Crossrefs

Cf. A000108, A007477, A307970, A307972.

Sequence in context: A051706 A292325 A367692 * A346073 A211696 A357533

Adjacent sequences: A307968 A307969 A307970 * A307972 A307973 A307974

Keyword

nonn

Author

Ilya Gutkovskiy, May 08 2019

Status

approved