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A128611
Number of Z-convex polyominoes with semiperimeter n.
1
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
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
Sequence in context: A224066 A150646 A364145 * A061539 A232970 A116078
KEYWORD
nonn
AUTHOR
Ralf Stephan, May 08 2007
STATUS
approved