A005435 Number of column-convex polyominoes with perimeter 2n+2. (Formerly M1779)

1, 2, 7, 28, 122, 558, 2641, 12822, 63501, 319554, 1629321, 8399092, 43701735, 229211236, 1210561517, 6432491192, 34364148528, 184463064936, 994430028087, 5381653402890, 29226425965907, 159227245772460, 870004781620093, 4766330416567254, 26176585256712224

Offset

1,2

References

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.

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..400

R. Brak, A. J. Guttmann and I. G. Enting, Exact solution of the row-convex perimeter generating function, J. Phys. A 23 (1990), 2319-2326.

M.-P. Delest, Generating functions for column-convex polyominoes, J. Combin. Theory Ser. A 48 (1988), no. 1, 12-31.

E. Duchi and S. Rinaldi, An object grammar for column-convex polyominoes, Annals of Combinatorics, 8 (2004), 27-36.

S. Feretic, A new way of counting the column-convex polyominoes by perimeter, Discrete Math., 180, 1998, 173-184.

Svjetlan Feretić, The perimeter generating function for nondirected diagonally convex polyominoes, arXiv:1907.09409 [math.CO], 2019. See Eq. (28).

Formula

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

Example

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.

Maple

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

Mathematica

$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 *)

Crossrefs

Cf. A006027, A269228.

Sequence in context: A150660 A150661 A269228 * A291091 A215973 A143927

Adjacent sequences: A005432 A005433 A005434 * A005436 A005437 A005438

Keyword

nonn,nice

Author

Simon Plouffe

Extensions

Corrected by Simon Plouffe.

More terms from Emeric Deutsch, May 13 2006

Status

approved