A075126 Number of bargraphs of site-perimeter n.

0, 0, 0, 1, 0, 2, 2, 4, 8, 14, 26, 52, 97, 193, 377, 752, 1500, 3022, 6107, 12429, 25365, 52042, 107090, 221235, 458316, 952439, 1984262, 4144601, 8676232, 18202536, 38263456, 80585694, 170013976, 359277183, 760400787, 1611721413, 3420839200, 7270098883

Offset: 1

Author

Andrew Rechnitzer (a.rechnitzer(AT)ms.unimelb.edu.au), Sep 09 2002

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Crossrefs

Cf. A075125.

Row sums of A274207.

Programs

Maple

b:= proc(n, y, t, w) option remember; `if`(n<0, 0, `if`(n=0, (1-t),
     `if`(t<0, 0, b(n-`if`(w>0 or t=0, 1, 2), y+1, 1, max(0, w-1)))+
     `if`(t>0 or y<2, 0, b(n, y-1, -1, `if`(t=0, 1, w+1)))+
     `if`(y<1, 0, b(n-`if`(t<0, 1, 2), y, 0, `if`(t<0, w, 0)))))
    end:
a:= n-> b(n, 0, 1, 0):
seq(a(n), n=1..45);  # Alois P. Heinz, Jun 05 2016

Mathematica

b[n_, y_, t_, w_] := b[n, y, t, w] = If[n<0, 0, If[n==0, 1-t, If[t<0, 0, b[n - If[w>0 || t==0, 1, 2], y+1, 1, Max[0, w-1]]] + If[t>0 || y<2, 0, b[n, y-1, -1, If[t==0, 1, w+1]]] + If[y<1, 0, b[n - If[t<0, 1, 2], y, 0, If[t<0, w, 0]]]]]; a[n_] := b[n, 0, 1, 0]; Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz *)

Formula

The generating function is known in closed form. Closed, but complicated. It is proved to be non-D-finite.

Extensions

Offset corrected by Alois P. Heinz, Jun 04 2016

a(31)-a(38) from Alois P. Heinz, Jun 05 2016