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.

0, 1, 1, 3, 9, 27, 117, 459, 2421, 11979, 74421, 443979, 3184821, 22216779, 180996021, 1444706379, 13186615221, 118495279179, 1198323664821, 11969865775179, 132880218064821, 1460470704175179, 17659740362704821

Offset

1,4

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.

Links

Table of n, a(n) for n=1..23.

Formula

Recurrence relation: a(n)=floor((n-1)/2)*a(n-1)+floor((n+1)/2)*a(n-2); a(1)=0, a(2)=1.

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[1]:=0: a[2]:=1: for n from 3 to 26 do a[n]:=floor((n-1)/2)*a[n-1]+floor((n+1)/2)*a[n-2] od: seq(a[n], n=1..26);

Crossrefs

Cf. A121745, A121749.

Sequence in context: A148929 A148930 A230951 * A323927 A146151 A306681

Adjacent sequences: A121743 A121744 A121745 * A121747 A121748 A121749

Keyword

nonn

Author

Emeric Deutsch, Aug 20 2006

Status

approved