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A360419
a(n) = the number of U-frame polyominoes with n cells, reduced for symmetry.
4
0, 0, 0, 0, 1, 2, 5, 9, 16, 24, 37, 50, 71, 93, 121, 151, 192, 231, 285, 338, 398, 470, 548, 626, 723, 827, 924, 1056, 1175, 1314, 1454, 1629, 1763, 1985, 2138, 2356, 2540, 2820, 2976, 3305, 3491, 3834, 4039, 4441, 4613, 5103, 5291, 5775, 5999, 6572
OFFSET
1,6
COMMENTS
A U-frame polyomino has a perimeter that forms a self-avoiding polygon such that as you traverse the perimeter counterclockwise you encounter turns in the order LLLLLLRR.
LINKS
FORMULA
G.f.: Sum_{k>=1} (x^k/(1 - x^k)) * (B(k+1, x)^2 + B(k+1, x^2))/2 where B(k, x) = Sum_{j>=k} x^j/(1 - x^j). - Andrew Howroyd, Feb 07 2023
EXAMPLE
a(5)=1 because of:
OO
O
OO
The a(7) = 5 polyominoes are:
O
O O O O
O O O O O OO O O O O
OOO OOO OOOO OOOO OOOOO
PROG
(PARI) B(n, k, x) = sum(j=k, n, x^j/(1 - x^j), O(x*x^n))
seq(n) = Vec(sum(k=1, (n-2)\3, x^k*(B(n-k, k+1, x)^2 + B((n-k)\2, k+1, x^2))/(1-x^k), O(x*x^n))/2, -n) \\ Andrew Howroyd, Feb 07 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
John Mason, Feb 06 2023
STATUS
approved