A273900 Number of columns of length 1 in all bargraphs of semiperimeter n (n>=2).

1, 2, 5, 14, 40, 116, 341, 1014, 3045, 9222, 28137, 86408, 266887, 828560, 2584111, 8092646, 25438494, 80235386, 253854855, 805447478, 2562252423, 8170557076, 26112495767, 83626191936, 268331079046, 862537758650, 2777237155053, 8956318767652, 28925845302365

Offset

2,2

Links

Alois P. Heinz, Table of n, a(n) for n = 2..1000

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Formula

G.f.: g(z)=(1-z)*(1-3z+z^2-z^3-(1-z)Q)/(2z^2), where Q = sqrt((1-z)(1-3z-z^2-z^3)).

a(n) = Sum(k*A273899(n,k), k>=1).

D-finite with recurrence (n+2)*a(n) -4*n*a(n-1) +2*(n-4)*a(n-2) -2*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Jul 24 2022

Example

a(4) = 5 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3]  and the corresponding pictures give the values 3, 1, 1, 0, 0 for the number of columns of length 1.

Maple

g:=(1/2)*(1-z)*(1-3*z+z^2-z^3-(1-z)*Q)/z^2: Q:=sqrt((1-z)*(1-3*z-z^2-z^3)): gser:= series(g, z=0, 40): seq(coeff(gser, z, n), n=2..35);
# Alternative:
a:= proc(n) option remember; `if`(n<5, [0, 1, 2, 5][n], (
      4*n*a(n-1)-2*(n-4)*a(n-2)+2*a(n-3)-(n-6)*a(n-4))/(n+2))
    end:
seq(a(n), n=2..35);  # Alois P. Heinz, Jun 07 2016

Mathematica

a[n_] := a[n] = If[n<5, {0, 1, 2, 5}[[n]], (4*n*a[n-1] - 2*(n-4)*a[n-2] + 2*a[n-3] - (n-6)*a[n-4])/(n+2)]; Table[a[n], {n, 2, 35}] (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

Crossrefs

Cf. A082582, A273899.

Sequence in context: A329275 A036908 A293346 * A126220 A136304 A190254

Adjacent sequences: A273897 A273898 A273899 * A273901 A273902 A273903

Keyword

nonn

Author

Emeric Deutsch, Jun 07 2016

Status

approved