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A005436 Number of convex polygons of perimeter 2n on square lattice.
(Formerly M1778)
9
1, 2, 7, 28, 120, 528, 2344, 10416, 46160, 203680, 894312, 3907056, 16986352, 73512288, 316786960, 1359763168, 5815457184, 24788842304, 105340982248, 446389242480, 1886695382192, 7955156287456, 33468262290096, 140516110684832, 588832418973280, 2463133441338048 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Or, a(n) = number of convex polyominoes of perimeter 2n. - David Callan, Jul 25 2008

REFERENCES

K. Y. Lin and S. J. Chang, J. Phys. A: Math. Gen., 21 (1988), 2635-2642.

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

LINKS

Robert Israel, Table of n, a(n) for n = 2..1650 (n = 2..105 from I. Jensen)

Peter Balazs, Generation and Empirical Investigation of hv-Convex Discrete Sets, in Image Analysis, Lecture Notes in Computer Science, Volume 4522/2007, Springer-Verlag. [From N. J. A. Sloane, Jul 09 2009]

D. Battaglino, J. M. Fedou, S. Rinaldi and S. Socci, The number of k-parallelogram polyominoes, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1143-1154.

A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining convex permutominoes, J. Int. Seq. 10 (2007) # 07.9.7.

Adrien Boussicault, P. Laborde-Zubieta, Periodic Parallelogram Polyominoes, arXiv preprint arXiv:1611.03766 [math.CO], 2016.

M.-P. Delest and G. Viennot, Algebraic languages and polyominoes enumeration, Theoretical Computer Sci., 34 (1984), 169-206.

F. Disanto, A. Frosini, R. Pinzani and S. Rinaldi, A closed formula for the number of convex permutominoes, arXiv:math/0702550 [math.CO], 2007.

Filippo Disanto, Andrea Frosini, Simone Rinaldi, Renzo Pinzani, The Combinatorics of Convex Permutominoes, Southeast Asian Bulletin of Mathematics (2008) 32: 883-912.

E. Duchi, S. Rinaldi and G. Schaeffer, The number of Z-convex polyominoes, arXiv:math/0602124 [math.CO], 2006.

I. G. Enting and A. J. Guttmann, Area-weighted moments of convex polygons on the square lattice, J. Phys. A 22 (1989), 2639-2642. See Eq. (4).

I. G. Enting and A. J. Guttmann, On the area of square lattice polygons, J. Statist. Phys., 58 (1990), 475-484. See p. 477.

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.

I. Jensen, More terms

Anne Micheli and Dominique Rossin, Counting k-Convex Polyominoes, Electron. J. Combin., Volume 20, Issue 2 (2013), #P56.

Eric Weisstein's World of Mathematics, Convex Polyomino

V. M. Zhuravlev, Horizontally-convex polyiamonds and their generating functions, Mat. Pros. 17 (2013), 107-129 (in Russian).

FORMULA

a(n) = (3+2*n)*4^n/256 - (4*n-12)*C(2n-7,n-4) for n >= 4.

(2*n+11)*4^n-4*(2*n+1)*binomial(2*n,n) produces the terms (except the first two) with a different offset. - N. J. A. Sloane, Oct 14 2017

G.f.: x^2*(1-6*x+11*x^2-4*x^3)/(1-4*x)^2-4*x^4*(1-4*x)^(-3/2). - Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003

MAPLE

t1:=x^2*( (1-6*x+11*x^2-4*x^3)/(1-4*x)^2 - 4*x^2/(1-4*x)^(3/2));

series(t1, x, 40);

gfun:-seriestolist(%); # N. J. A. Sloane, Aug 02 2015

MATHEMATICA

Join[{1, 2}, Table[(2 n + 11) 4^n - 4 (2 n + 1) Binomial[2 n, n], {n, 0, 25}]] (* Vincenzo Librandi, Jun 25 2015 *)

PROG

(MAGMA) [1, 2] cat [4^n*(2*n+11)-4*(2*n+1)*Binomial(2*n, n): n in [0..25]]; // Vincenzo Librandi, Jun 25 2015

CROSSREFS

a(n) = A005768(n) + A005769(n) + A005770(n).

Cf. A260346.

Sequence in context: A150653 A150654 A150655 * A150656 A150657 A150658

Adjacent sequences:  A005433 A005434 A005435 * A005437 A005438 A005439

KEYWORD

nonn

AUTHOR

Simon Plouffe and N. J. A. Sloane.

EXTENSIONS

First formula corrected by Robert Israel, Apr 04 2016

STATUS

approved

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Last modified March 22 04:32 EDT 2019. Contains 321406 sequences. (Running on oeis4.)