A186537 G.f.: Sum( x^k/(1-2*x+x^k), k=1..oo).

0, 1, 2, 4, 7, 12, 20, 34, 58, 101, 178, 318, 574, 1046, 1920, 3548, 6593, 12312, 23092, 43480, 82154, 155716, 295984, 564050, 1077400, 2062311, 3955186, 7598756, 14622318, 28179338, 54379520, 105071498, 203254164, 393607534, 763001000, 1480458656, 2875091021, 5588152920, 10869906136

Offset: 0

Author

N. J. A. Sloane, Feb 23 2011

Keywords

• nonn

Comments

This arose while studying the properties of A079500.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 257 terms from N. J. A. Sloane)

Crossrefs

First differences give A079500.

Programs

Maple

add( x^k/(1-2*x+x^k), k=1..61); series(%,x,60); seriestolist(%);
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 1,
      `if`(m=0, add(b(n-j, j), j=1..n),
      add(b(n-j, min(n-j, m)), j=1..min(n, m))))
    end:
a:= proc(n) a(n):= `if`(n=0, 0, b(n-1, 0)+a(n-1)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 01 2014

Mathematica

b[n_, m_] := b[n, m] = If[n == 0, 1, If[m == 0, Sum[b[n-j, j], {j, 1, n}], Sum[b[n-j, Min[n-j, m]], {j, 1, Min[n, m]}]]]; a[n_] := If[n == 0, 0, b[n-1, 0] + a[n-1]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 05 2014, after Alois P. Heinz *)

Formula

G.f.: -(1+x^2+ 1/(x-1) )/(1-x)*( 1 + x*(x-1)^3*(1-x+x^3)/( Q(0)- x*(x-1)^3*(1-x+x^3)) ), where Q(k) = (x+1)*(2*x-1)*(1-x)^2 + x^(k+2)*(x+x^2+x^3-2*x^4-1 - x^(k+3) + x^(k+5)) - x*(-1+2*x-x^(k+3))*(1-2*x+x^2+x^(k+4)-x^(k+5))*(-1+4*x-5*x^2+2*x^3 - x^(k+2)- x^(k+5) + 2*x^(k+3) - x^(2*k+5) + x^(2*k+6))/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 14 2013