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%I M1778 #95 Jan 13 2024 04:53:01
%S 1,2,7,28,120,528,2344,10416,46160,203680,894312,3907056,16986352,
%T 73512288,316786960,1359763168,5815457184,24788842304,105340982248,
%U 446389242480,1886695382192,7955156287456,33468262290096,140516110684832,588832418973280,2463133441338048
%N Number of convex polygons of perimeter 2n on square lattice.
%C Or, a(n) = number of convex polyominoes of perimeter 2n. - _David Callan_, Jul 25 2008
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert Israel, <a href="/A005436/b005436.txt">Table of n, a(n) for n = 2..1650</a> (n = 2..105 from I. Jensen)
%H Peter Balazs, <a href="http://dx.doi.org/10.1007/978-3-540-73040-8_35">Generation and Empirical Investigation of hv-Convex Discrete Sets</a>, in Image Analysis, Lecture Notes in Computer Science, Volume 4522/2007, Springer-Verlag. [From _N. J. A. Sloane_, Jul 09 2009]
%H D. Battaglino, J. M. Fedou, S. Rinaldi and S. Socci, <a href="https://doi.org/10.46298/dmtcs.2370">The number of k-parallelogram polyominoes</a>, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1143-1154.
%H A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rinaldi/rinaldi5.html">Permutations defining convex permutominoes</a>, J. Int. Seq. 10 (2007) # 07.9.7.
%H Adrien Boussicault and P. Laborde-Zubieta, <a href="https://arxiv.org/abs/1611.03766">Periodic Parallelogram Polyominoes</a>, arXiv preprint arXiv:1611.03766 [math.CO], 2016.
%H Adrien Boussicault, Simone Rinaldi, and Samanta Socci, <a href="https://arxiv.org/abs/1501.00872">The number of directed k-convex polyominoes</a>, arXiv preprint arXiv:1501.00872 [math.CO], 2015; Discrete Math., 343 (2020), #111731, 22 pages.
%H Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, <a href="https://arxiv.org/abs/1903.01095">The Number of Convex Polyominoes with Given Height and Width</a>, arXiv:1903.01095 [math.CO], 2019.
%H M.-P. Delest and G. Viennot, <a href="http://dx.doi.org/10.1016/0304-3975(84)90116-6">Algebraic languages and polyominoes enumeration</a>, Theoretical Computer Sci., 34 (1984), 169-206.
%H F. Disanto, A. Frosini, R. Pinzani and S. Rinaldi, <a href="http://arxiv.org/abs/math/0702550">A closed formula for the number of convex permutominoes</a>, arXiv:math/0702550 [math.CO], 2007.
%H Filippo Disanto, Andrea Frosini, Simone Rinaldi, and Renzo Pinzani, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_200805&filename=The%20Combinatorics%20of%20Convex%20Permutominoes.pdf">The Combinatorics of Convex Permutominoes</a>, Southeast Asian Bulletin of Mathematics (2008) 32: 883-912.
%H E. Duchi, S. Rinaldi and G. Schaeffer, <a href="http://arXiv.org/abs/math.CO/0602124">The number of Z-convex polyominoes</a>, arXiv:math/0602124 [math.CO], 2006.
%H I. G. Enting and A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/22/14/013">Area-weighted moments of convex polygons on the square lattice</a>, J. Phys. A 22 (1989), 2639-2642. See Eq. (4).
%H I. G. Enting and A. J. Guttmann, <a href="http://dx.doi.org/10.1007/BF01112757">On the area of square lattice polygons</a>, J. Statist. Phys., 58 (1990), 475-484. See p. 477.
%H A. J. Guttmann and I. G. Enting, <a href="http://dx.doi.org/10.1088/0305-4470/21/8/007">The number of convex polygons on the square and honeycomb lattices</a>, J. Phys. A 21 (1988), L467-L474.
%H I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/polygons/Polygons_ser.html">More terms</a>
%H K. Y. Lin and S. J. Chang, <a href="https://doi.org/10.1088/0305-4470/21/11/020">Rigorous results for the number of convex polygons on the square and honeycomb lattices</a>J. Phys. A: Math. Gen., 21 (1988), 2635-2642.
%H Anne Micheli and Dominique Rossin, <a href="https://doi.org/10.37236/3435">Counting k-Convex Polyominoes</a>, Electron. J. Combin., Volume 20, Issue 2 (2013), #P56.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConvexPolyomino.html">Convex Polyomino</a>
%H V. M. Zhuravlev, <a href="http://www.mccme.ru/free-books/matpros/mph.pdf">Horizontally-convex polyiamonds and their generating functions</a>, Mat. Pros. 17 (2013), 107-129 (in Russian).
%F a(n) = (2*n + 3)*4^(n-4) - 4*(n-3)*C(2*n-7, n-4) for n >= 4. - Corrected by _Robert Israel_, Apr 04 2016
%F a(n) = A005768(n) + A005769(n) + A005770(n).
%F a(n) = (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
%F 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
%F a(n) = (2*(8*n^2 -106*n +303)*a(n-1) - 8*(2*n-9)*(2*n-17)*a(n-2))/((n-4)*(2*n- 19)), with a(2) = 1, a(3) = 2, a(4) = 7, a(4) = 28. - _G. C. Greubel_, Nov 20 2022
%p t1:=x^2*( (1-6*x+11*x^2-4*x^3)/(1-4*x)^2 - 4*x^2/(1-4*x)^(3/2));
%p series(t1,x,40);
%p gfun:-seriestolist(%); # _N. J. A. Sloane_, Aug 02 2015
%t 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 *)
%o (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
%o (SageMath)
%o def A005436(n): return (2*n+3)*4^(n-4) -4*(n-3)*binomial(2*n-7, n-4) + (9/16)*int(n==2) - (1/4)*int(n==3)
%o [A005436(n) for n in range(2,40)] # _G. C. Greubel_, Nov 20 2022
%Y Cf. A005768, A005769, A005770, A093118, A260346, A324009.
%K nonn
%O 2,2
%A _Simon Plouffe_ and _N. J. A. Sloane_
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