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A121746
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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.
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2
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0, 1, 1, 3, 9, 27, 117, 459, 2421, 11979, 74421, 443979, 3184821, 22216779, 180996021, 1444706379, 13186615221, 118495279179, 1198323664821, 11969865775179, 132880218064821, 1460470704175179, 17659740362704821
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OFFSET
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1,4
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COMMENTS
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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