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Number of column-convex polyominoes with perimeter 2n+2.
(Formerly M1779)
9

%I M1779 #34 Mar 28 2021 01:40:27

%S 1,2,7,28,122,558,2641,12822,63501,319554,1629321,8399092,43701735,

%T 229211236,1210561517,6432491192,34364148528,184463064936,

%U 994430028087,5381653402890,29226425965907,159227245772460,870004781620093,4766330416567254,26176585256712224

%N Number of column-convex polyominoes with perimeter 2n+2.

%D S. Feretic and D. Svrtan, On the number of column-convex polyominoes with given perimeter and number of columns, Proc. 5th Conf. Formal Power Series and Algebraic Combinatorics, Florence, 1993, pp. 201-214.

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

%H Alois P. Heinz, <a href="/A005435/b005435.txt">Table of n, a(n) for n = 1..400</a>

%H R. Brak, A. J. Guttmann and I. G. Enting, <a href="http://dx.doi.org/10.1088/0305-4470/23/12/016">Exact solution of the row-convex perimeter generating function</a>, J. Phys. A 23 (1990), 2319-2326.

%H M.-P. Delest, <a href="http://dx.doi.org/10.1016/0097-3165(88)90071-4">Generating functions for column-convex polyominoes</a>, J. Combin. Theory Ser. A 48 (1988), no. 1, 12-31.

%H E. Duchi and S. Rinaldi, <a href="http://dx.doi.org/10.1007/s00026-004-0202-x">An object grammar for column-convex polyominoes</a>, Annals of Combinatorics, 8 (2004), 27-36.

%H S. Feretic, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00114-3">A new way of counting the column-convex polyominoes by perimeter</a>, Discrete Math., 180, 1998, 173-184.

%H Svjetlan Feretić, <a href="https://arxiv.org/abs/1907.09409">The perimeter generating function for nondirected diagonally convex polyominoes</a>, arXiv:1907.09409 [math.CO], 2019. See Eq. (28).

%F See the g.f. in the Maple program (taken from the Brak et al. paper). It has been given previously, in a different form, in the Delest paper (p. 29). - _Emeric Deutsch_, May 13 2006

%e a(3)=7 because we have: the 2 X 2 square, the 3 X 1 and 1 X 3 rectangles and the four polyominoes obtained by removing any of the four cells of the 2 X 2 square.

%p assume(y,positive): G:=((y^2 - 1)*( - 21 + 47*y^2 - 35*y^4 + 5*y^6) - 3*(y^2 - 1)^2*(1 + y^2)*sqrt(1 - 6*y^2 + y^4) - 9*sqrt(2)*(y^2 - 1)^2*sqrt((y^2 - 1)^2*(1 + y^2) - (y^2 - 1)*(1 + y^2)*sqrt(1 - 6*y^2 + y^4)) - sqrt(2)*y*(y^2 - 1)*(1 + y^2)*sqrt((y^2 - 1)^2*(1 + y^2) + (y^2 - 1)*(1 + y^2)*sqrt(1 - 6*y^2 + y^4)))/(72 - 152*y^2 + 92*y^4 - 8*y^6): Gser:=series(G,y=0,70): seq(coeff(Gser,y^(2*n+2)),n=1..31); # _Emeric Deutsch_, May 13 2006

%t $Assumptions = (y > 0); terms = 25; ((y^2 - 1)*(-21 + 47*y^2 - 35*y^4 + 5*y^6) - 3*(y^2 - 1)^2*(1 + y^2)*Sqrt[1 - 6*y^2 + y^4] - 9*Sqrt[2]*(y^2 - 1)^2*Sqrt[(y^2 - 1)^2*(1 + y^2) - (y^2 - 1)*(1 + y^2)*Sqrt[1 - 6*y^2 + y^4]] - Sqrt[2]*y*(y^2 - 1)*(1 + y^2)*Sqrt[(y^2 - 1)^2*(1 + y^2) + (y^2 - 1)*(1 + y^2)*Sqrt[1 - 6*y^2 + y^4]])/(72 - 152*y^2 + 92*y^4 - 8*y^6) + O[y]^(2 terms + 3) // Normal // Simplify // CoefficientList[#, y^2]& // #[[3 ;; terms + 2]]& (* _Jean-François Alcover_, May 15 2017, translated from Maple *)

%Y Cf. A006027, A269228.

%K nonn,nice

%O 1,2

%A _Simon Plouffe_

%E Corrected by _Simon Plouffe_.

%E More terms from _Emeric Deutsch_, May 13 2006