A346047 a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=1..n-3} a(k) * a(n-k-3).

1, 1, 1, 0, 1, 2, 2, 2, 4, 8, 10, 13, 24, 42, 61, 90, 156, 265, 410, 646, 1093, 1834, 2948, 4789, 8050, 13475, 22129, 36570, 61435, 103039, 171384, 286156, 481691, 810502, 1359194, 2284789, 3856974, 6512001, 10982193, 18550116, 31406597, 53194727, 90082902

Offset: 0

Author

Ilya Gutkovskiy, Jul 02 2021

Keywords

• nonn

Crossrefs

Cf. A025250, A050253, A307970, A343304, A346048, A346049.

Programs

Mathematica

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Sum[a[k] a[n - k - 3], {k, 1, n - 3}]; Table[a[n], {n, 0, 42}]
nmax = 42; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

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

a(n) ~ sqrt((3 - 3*r^3 - 4*r^4 - 2*r^5)/(8*Pi)) / (n^(3/2) * r^(n+3)), where r = 0.5701490701528437821032230160646366013461622472504286581627... is the root of the equation 1 - 2*r^3 - 4*r^4 - 4*r^5 + r^6 = 0. - Vaclav Kotesovec, Jul 03 2021