A307970 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3*A(x)^2.

1, 1, 1, 1, 2, 3, 4, 7, 12, 19, 32, 56, 96, 165, 290, 512, 902, 1601, 2862, 5124, 9198, 16585, 29990, 54336, 98702, 179742, 327942, 599432, 1097756, 2013737, 3699596, 6806866, 12541518, 23137270, 42736850, 79031394, 146309968, 271142469, 502978944, 933921458, 1735634266

Offset: 0

Author

Ilya Gutkovskiy, May 08 2019

Keywords

• nonn

Comments

Shifts 3 places left when convolved with itself.

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 12*x^8 + 19*x^9 + 32*x^10 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A000108, A007477, A307971, A307972.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, 1,
      add(a(j)*a(n-3-j), j=0..n-3))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2019

Mathematica

terms = 40; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; a[0] = a[1] = a[2] = 1; Table[a[n], {n, 0, 40}]

Formula

G.f.: 1/(1 - x/(1 - x^3/(1 - x^3/(1 - x/(1 - x^3/(1 - x^3/(1 - x/(1 - x^3/(1 - x^3/(1 - ...)))))))))), a continued fraction.

Recurrence: a(n+3) = Sum_{k=0..n} a(k)*a(n-k).

a(n) ~ sqrt(3 + 4*r^4 + 8*r^5) / (4*sqrt(Pi)*n^(3/2)*r^(n+3)), where r = 0.51899425841331458784223152875297289010563957455264491744143... is the root of the equation 1 + r + r^2 = 1/(4*r^3). - Vaclav Kotesovec, Jul 03 2021