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

0,4

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.

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

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 *)

Crossrefs

Cf. A003480, A005436.

Sequence in context: A224066 A150646 A364145 * A061539 A232970 A116078

Adjacent sequences: A128608 A128609 A128610 * A128612 A128613 A128614

Keyword

nonn

Author

Ralf Stephan, May 08 2007

Status

approved