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

1,2

Comments

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

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.

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)

Example

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

Mathematica

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

Prog

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

Crossrefs

Main diagonal of A334552.

Cf. A268404.

Sequence in context: A327646 A244746 A110051 * A273023 A013187 A069639

Adjacent sequences: A334548 A334549 A334550 * A334552 A334553 A334554

Keyword

nonn

Author

Andrew Howroyd, Jun 06 2020

Status

approved