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A005436 Number of convex polygons of perimeter 2n on square lattice.
(Formerly M1778)
11
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

Boussicault, Adrien, Simone Rinaldi, and Samanta Socci. "The number of directed k-convex polyominoes." arXiv preprint arXiv:1501.00872 (2015). Discrete Math., 343 (2020), #111731, 22 pages.

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.

Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.

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 September 22 11:22 EDT 2020. Contains 337289 sequences. (Running on oeis4.)