OFFSET
0,5
COMMENTS
a(n) = A121745(n,0).
REFERENCES
E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29- 42.
FORMULA
Recurrence relation: a(n) = floor((n-1)/2)*a(n-1)+floor((n+1)/2)*a(n-2); a(0)=1, a(1)=0.
G.f.: Q(0)/(x*(1+x)) - 1/x, where Q(k)= 1 + x*(k+1)/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 23 2013
Let S=Sum_{n>=0} (1 + x*n + x)*x^(2*n)*(n!)^2, then g.f. A(x)=S/(x+x^2) - 1/x. - Sergei N. Gladkovskii, May 23 2013
D-finite with recurrence 4*a(n) +2*a(n-1) +(-n^2+n-2)*a(n-2) -n*(n-1)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes and only the vertical one consists only of columns of even length.
MAPLE
a[0]:=1: a[1]:=0: for n from 2 to 26 do
a[n]:=floor((n-1)/2)*a[n-1]+floor((n+1)/2)*a[n-2] od:
seq(a[n], n=0..26);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 20 2006
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Aug 19 2024
STATUS
approved