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A334551
Number of fixed polyominoes with 2n-1 cells and width and height both equal to n.
4
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 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
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Jun 06 2020
STATUS
approved