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

Author

John Mason, Feb 06 2023

Keywords

• nonn

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.

Examples

			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
		

Links

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

Crossrefs

Cf. A028247, A270060, A360420, A360421.

Programs

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

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