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A006743 Number of convex polygons of length 2n on honeycomb, or EG-convex polyominoes.
(Formerly M2255)
0
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 (list; graph; refs; listen; history; text; internal format)
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 A075856 A025520

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

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Last modified February 23 05:58 EST 2019. Contains 320411 sequences. (Running on oeis4.)