A332846 a(1) = 1; a(n+1) = Sum_{k=1..n} a(k) * ceiling(n/k).

1, 1, 3, 8, 20, 50, 121, 297, 716, 1739, 4198, 10157, 24513, 59246, 143006, 345381, 833792, 2013272, 4860337, 11734717, 28329772, 68396030, 165121957, 398644144, 962410246, 2323475153, 5609360573, 13542220814, 32693802921, 78929886033, 190553574988, 460037180829, 1110627936647

Offset: 1

Author

Ilya Gutkovskiy, Feb 26 2020

Keywords

• nonn

Crossrefs

Cf. A006590, A014668, A097919, A332490.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = Sum[a[k] Ceiling[(n - 1)/k], {k, 1, n - 1}]; Table[a[n], {n, 1, 33}]
a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] + Sum[a[d], {d, Divisors[k]}], {k, 1, n - 2}]; Table[a[n], {n, 1, 33}]
terms = 33; A[] = 0; Do[A[x] = x (1 + (1/(1 - x)) (A[x] + x Sum[A[x^k], {k, 1, terms}])) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * (A(x) + x * Sum_{k>=1} A(x^k))).

a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-2} (a(k) + Sum_{d|k} a(d)).

a(n) ~ c * (1 + sqrt(2))^n, where c = 0.2594006517235012546870541901936538347053403598092060748627156661727... - Vaclav Kotesovec, Mar 10 2020