A176528 a(2*n) = n*a(n); a(2*n+1) = n*a(n) + a(n+1), with a(1) = 1.

1, 1, 2, 2, 4, 6, 8, 8, 12, 20, 26, 36, 44, 56, 64, 64, 76, 108, 128, 200, 226, 286, 322, 432, 476, 572, 628, 784, 848, 960, 1024, 1024, 1100, 1292, 1400, 1944, 2072, 2432, 2632, 4000, 4226, 4746, 5032, 6292, 6614, 7406, 7838, 10368, 10844, 11900, 12472, 14872

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Formula

Let M = an infinite triangular matrix with (1, k, k, 0, 0, 0,...) in the k-th column stepped down twice from the previous column, for k>1.

A176528 = Lim_{n->inf} M^n, the left-shifted vector considered as a sequence.

Example

First few rows of the generating triangle M =
1;
1;
1, 1;
0, 2;
0, 2, 1;
0, 0, 3;
0, 0, 3, 1;
0, 0, 0, 4;
0, 0, 0, 4, 1;
...
Examples: a(10) = n * a(5) = 5 * 4 = 20.
a(11) = 5 * a(5) + a(6) = 5 * 4 + 6 = 26.

Maple

a:= proc(n) option remember; `if`(n=1, 1, (h->
      h*a(h)+`if`(n::odd, a(h+1), 0))(iquo(n, 2)))
    end:
seq(a(n), n=1..55);  # Alois P. Heinz, Apr 15 2021

Mathematica

a[n_] := a[n] = If[n == 1, 1, With[{h = Quotient[n, 2]}, h*a[h] + If[OddQ[n], a[h+1], 0]]];
Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

Prog

(PARI) a(n) = {if(n<=1, n==1, my(t=n\2); t*a(t) + if(n%2, a(t+1)))} \\ Andrew Howroyd, Apr 15 2021

Crossrefs

Sequence in context: A194695 A194693 A308841 * A108891 A147570 A049625

Adjacent sequences: A176525 A176526 A176527 * A176529 A176530 A176531

Keyword

nonn

Author

Gary W. Adamson, Apr 19 2010

Extensions

Terms a(28) and beyond from Andrew Howroyd, Apr 15 2021

Status

approved