A121302 - OEIS

A121302

Number of directed column-convex polyominoes having at least one 1-cell column.

1, 1, 4, 10, 28, 75, 202, 540, 1440, 3828, 10153, 26875, 71021, 187421, 494013, 1300844, 3422509, 8998118, 23642479, 62088032, 162978242, 427648023, 1121766397, 2941697012, 7712415568, 20215976824, 52981414253, 138831400836

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

a(n) = Fibonacci(2n-1) - A121469(n,0) (obviously, since A121469(n,k) is the number of directed column-convex polyominoes of area n having k 1-cell columns). Column 1 of A121301.

LINKS

Table of n, a(n) for n=1..28.

E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.

Index entries for linear recurrences with constant coefficients, signature (5,-6,-2,4,-1).

FORMULA

G.f.: z(1-z)(1-3z+2z^2)/[(1-3z+z^2)(1-2z-z^2+z^3)].

a(n) = A001519(n)-A077998(n-2), n>0. - R. J. Mathar, Jul 22 2022

EXAMPLE

a(3)=4 because, with the exception of the 3-cell column, all the other four directed column-convex polyominoes of area 3 have a 1-cell column.

MAPLE

G:=z*(1-z)*(1-3*z+2*z^2)/(1-3*z+z^2)/(1-2*z-z^2+z^3): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=1..32);

PROG

(PARI) Vec(z*(1-z)*(1-3*z+2*z^2)/((1-3*z+z^2)*(1-2*z-z^2+z^3)) + O(z^40)) \\ Michel Marcus, Feb 14 2016

CROSSREFS

Cf. A121469, A121301.