A334551 Number of fixed polyominoes with 2n-1 cells and width and height both equal to n.

1, 4, 25, 120, 497, 1924, 7265, 27288, 102745, 388692, 1477721, 5643064, 21632785, 83204260, 320932177, 1240939448, 4808642313, 18668848852, 72601081385, 282762109272, 1102772229313, 4306062994148, 16832791708257, 65867445819160, 257980829463017

Offset: 1

Author

Andrew Howroyd, Jun 06 2020

Keywords

• nonn

Comments

A polyomino with width and height equal to n must have at least 2n-1 cells.

Examples

			a(3) = 25. Up to rotation and reflection there are 6 possibilities:
       X       X   X         X         X     X
       X     X X   X X X   X X X   X X X   X X X
   X X X   X X     X       X       X         X
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500
• Andrew Conway, Enumerating 2D percolation series by the finite-lattice method: theory, J. Phys. A: Math. Gen., 28 (1995), 335-349. See Table 4.

Crossrefs

Main diagonal of A334552.

Cf. A268404.

Programs

Mathematica

Array[8Binomial[2(#-1),#-1]-3#^2+4#-8&,50] (* Paolo Xausa, Dec 21 2023 *)

PARI

a(n) = 8*binomial(2*(n-1), n-1) - 3*n^2 + 4*n - 8; \\ Peter J. Taylor, Dec 15 2020

Formula

a(n) = 2*binomial(2*(n-1), n-1) + 4*(n-2) + (n-2)^2*(2*n-5) + 2*Sum_{i=1..n-2} Sum_{j=1..n-2} ((n-2-i)*(n-2-j)+2)*binomial(i+j, i) for n > 1.

a(n) = 8*binomial(2*(n-1), n-1) - 3*n^2 + 4*n - 8. - Peter J. Taylor, Dec 15 2020

From Stefano Spezia, Sep 02 2022: (Start)

G.f.: 8*x/sqrt(1 - 4*x) - (8 - 17*x + 15*x^2)/(1 - x)^3.

a(n) ~ 2^(2*n+1)/sqrt(n*Pi). (End)