A320998 Number of pseudo-square convex polyominoes with semiperimeter n.

1, 12, 44, 142, 399, 1044, 2571, 6168, 14357, 32786, 73746, 163872, 360462, 786468, 1703949, 3670040, 7864353, 16777260, 35651579, 75497508, 159383591, 335544350, 704643087, 1476395064, 3087007733, 6442451004, 13421772816, 27917287460, 57982058547, 120259084318

Offset: 6

Author

N. J. A. Sloane, Oct 30 2018

Keywords

• nonn

Comments

The offset is not specified but appears to be 6.

Links

• Andrew Howroyd, Table of n, a(n) for n = 6..1000
• Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol. 13, (2006). See Section 4.2.

Crossrefs

Cf. A045618, A320999.

Programs

Maple

seq(coeff(series(2*x^6/((1-x)^2*(1-2*x)^2)-add(k*x^(3*(k+1))/(1-x^(k+1))^2,k=1..ceil(n/3)),x,n+1), x, n), n = 6 .. 35); # Muniru A Asiru, Oct 31 2018

Mathematica

seq[n_] := 2*x^6/((1 - x)^2*(1 - 2*x)^2) - Sum[k*x^(3*(k + 1))/(1 - x^(k + 1))^2 + O[x]^(6 + n), {k, 1, Ceiling[n/3]}] // CoefficientList[#, x]& // Drop[#, 6]&;
seq[30] (* Jean-François Alcover, Sep 07 2019, from PARI *)

PARI

seq(n)={Vec(2*x^6/((1-x)^2*(1-2*x)^2) - sum(k=1, ceil(n/3), k*x^(3*(k+1))/(1-x^(k+1))^2 + O(x^(6+n))))} \\ Andrew Howroyd, Oct 31 2018

Formula

a(n) = 2*A045618(6+n) - A320999(n). - Andrew Howroyd, Oct 31 2018

Extensions

Terms a(16) and beyond from Andrew Howroyd, Oct 31 2018