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A121746
Number of deco polyominoes of height n, consisting only of columns of even length. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
2
1, 0, 1, 1, 3, 9, 27, 117, 459, 2421, 11979, 74421, 443979, 3184821, 22216779, 180996021, 1444706379, 13186615221, 118495279179, 1198323664821, 11969865775179, 132880218064821, 1460470704175179, 17659740362704821, 211782792439855179, 2770970133993424821
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
Sequence in context: A148929 A148930 A230951 * A323927 A146151 A306681
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 20 2006
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Aug 19 2024
STATUS
approved