A128611 Number of Z-convex polyominoes with semiperimeter n.

0, 0, 1, 2, 7, 28, 116, 484, 2022, 8448, 35290, 147376, 615228, 2567060, 10704976, 44611804, 185780308, 773060804, 3214225836, 13352979316, 55426067494, 229870371888, 952548347122, 3943943111920, 16316243701350, 67447113649312, 278592165886198, 1149863118820584, 4742473257979906, 19545876370622104, 80502059920697442

Offset: 0

Author

Ralf Stephan, May 08 2007

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..1658
• Adrien Boussicault, Simone Rinaldi, and Samanta Socci, The number of directed k-convex polyominoes, arXiv preprint arXiv:1501.00872 [math.CO], 2015; Discrete Math., 343 (2020), #111731, 22 pages. See page 2.
• E. Duchi, S. Rinaldi and G. Schaeffer, The number of Z-convex polyominoes, arXiv:math/0602124 [math.CO], 2006.

Crossrefs

Cf. A003480, A005436.

Programs

Maple

d:=(1-2*t-sqrt(1-4*t))/2:
t1:=
2*t^4*(1-2*t)^2*d/( (1-4*t)^2*(1-3*t)*(1-t) )
+ t^2*(1-6*t+10*t^2-2*t^3-t^4)/( (1-4*t)*(1-3*t)*(1-t) ):
series(t1,t,120):
seriestolist(%); # N. J. A. Sloane, Oct 02 2011

Mathematica

gf = 2 t^4 (1-2t)^2 d/((1-4t)^2 (1-3t)(1-t)) + t^2 (1-6t+10t^2-2t^3-t^4)/ ((1-4t)(1-3t)(1-t)) /. d -> (1-2t-Sqrt[1-4t])/2;
CoefficientList[gf + O[t]^31, t] (* Jean-François Alcover, Aug 17 2018 *)

Formula

The Duchi paper has a g.f.

Asymptotically, a(n) ~ n/24 * 4^n.

G.f.: Let d:=(1-2*t-sqrt(1-4*t))/2; then g.f. is 2*t^4*(1-2*t)^2*d/( (1-4*t)^2*(1-3*t)*(1-t) ) + t^2*(1-6*t+10*t^2-2*t^3-t^4)/( (1-4*t)*(1-3*t)*(1-t) ). - N. J. A. Sloane, Oct 02 2011

(-960+384*n)*a(n)+(1760-992*n)*a(n+1)+(-924+984*n)*a(2+n)+(64-490*n)*a(n+3)+(82+131*n)*a(n+4)+(-24-18*n)*a(n+5)+(2+n)*a(n+6), a(0) = 0. - Robert Israel, Aug 17 2018