%I #15 Feb 23 2023 13:21:30

%S 0,0,0,0,1,2,5,9,16,24,37,50,71,93,121,151,192,231,285,338,398,470,

%T 548,626,723,827,924,1056,1175,1314,1454,1629,1763,1985,2138,2356,

%U 2540,2820,2976,3305,3491,3834,4039,4441,4613,5103,5291,5775,5999,6572

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

%C 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.

%H Andrew Howroyd, <a href="/A360419/b360419.txt">Table of n, a(n) for n = 1..1000</a>

%F 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

%e a(5)=1 because of:

%e OO

%e O

%e OO

%e The a(7) = 5 polyominoes are:

%e O

%e O O O O

%e O O O O O OO O O O O

%e OOO OOO OOOO OOOO OOOOO

%o (PARI) B(n,k,x) = sum(j=k, n, x^j/(1 - x^j), O(x*x^n))

%o 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

%Y Cf. A028247, A270060, A360420, A360421.

%K nonn

%O 1,6

%A _John Mason_, Feb 06 2023