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%I M2255 #25 Aug 28 2018 12:07:32
%S 1,0,3,2,10,14,40,74,176,358,798,1670,3626,7638,16366,34462,73230,
%T 153830,324896,680514,1430336,2987310,6253712,13025954,27176052,
%U 56465878,117458820,243507250,505239264,1045301486
%N Number of convex polygons of length 2n on honeycomb, or EG-convex polyominoes.
%D 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.
%D 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.
%D 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.
%D 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%F 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_
%F 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
%t 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 *)
%K nonn,easy,nice
%O 3,3
%A _Simon Plouffe_, _N. J. A. Sloane_
%E Additional references from _Fouad IBN MAJDOUB HASSANI_, Feb 28 2000
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