A360419 a(n) = the number of U-frame polyominoes with n cells, reduced for symmetry.

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

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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

Cf. A028247, A270060, A360420, A360421.

Sequence in context: A345140 A072829 A169740 * A282044 A138226 A175287

Adjacent sequences: A360416 A360417 A360418 * A360420 A360421 A360422

Keyword

nonn

Author

John Mason, Feb 06 2023

Status

approved