A006743 Number of convex polygons of length 2n on honeycomb, or EG-convex polyominoes. (Formerly M2255)

1, 0, 3, 2, 10, 14, 40, 74, 176, 358, 798, 1670, 3626, 7638, 16366, 34462, 73230, 153830, 324896, 680514, 1430336, 2987310, 6253712, 13025954, 27176052, 56465878, 117458820, 243507250, 505239264, 1045301486

Offset

3,3

References

Alain Denise, Christoph Durr and Fouad Ibn-Majdoub-Hassani. Enumeration et generation aleatoire de polyominos convexes en reseau hexagonal (French) [enumeration and random generation of convex polyominoes in the honeycomb lattice]. In Proceedings of 9th Conference on Formal Power Series and Algebraic Combinatorics, pages 222-234, 1997.

A. J. Guttmann and I. G. Enting, The number of convex polygons on the square and honeycomb lattices, J. Phys. A 21 (1988), L467-L474.

Fouad Ibn-Majdoub-Hassani, Combinatoire de polyominos et des tableaux decales oscillants, These de Doctorat, Laboratoire de Recherche en Informatique, Universite Paris-Sud XI, France.

K. Y. Lin, S. J. Chang, rigorous results for the number of convex polygons on the square and honeycomb lattices, J. Phys A: Math. Gen. 21 (1988) 2635

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=3..32.

Formula

G.f.: (1-2*x+x^2-x^4-x^2*sqrt(1-4*x^2))/((1+x)^2*(1-2*x)^2). - Paul Zimmermann

For n>=10, (n-5) a(n) = 2(n-5) a(n-1) + (7n-47) a(n-2) + 12(7-n) a(n-3) + 4(29-4n) a(n-4) + 16(n-8) a(n-5) + 16(n-8) a(n-6). This follows from the differential equation (2-2x-12x^2+12x^3+4x^4+16x^6) g(x) + (-x+2x^2+7x^3-12x^4-16x^5+16x^6+16x^7) g'(x) = 2-2x-12x^2+16x^3-2x^4-4x^6 satisfied by the g.f. sum_n>=0 a(n+3) x^n = (1-2x+x^2-x^4-x^2 sqrt(1-4x^2))/((1+x)^2 (1-2x)^2). - Dean Hickerson, Oct 26 2005

Mathematica

CoefficientList[ Series[(1 - 2x + x^2 - x^4 - x^2*Sqrt[1 - 4*x^2])/(1 + x)^2/(1 - 2*x)^2, {x, 0, 29}], x] (* Robert G. Wilson v, Oct 15 2005 *)

Crossrefs

Sequence in context: A226442 A300374 A256063 * A091811 A327809 A075856

Adjacent sequences: A006740 A006741 A006742 * A006744 A006745 A006746

Keyword

nonn,easy,nice

Author

Simon Plouffe, N. J. A. Sloane

Extensions

Additional references from Fouad IBN MAJDOUB HASSANI, Feb 28 2000

Status

approved