A181328 Number of columns with an even sum in all 2-compositions of n. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

0, 0, 3, 12, 59, 248, 1024, 4080, 15948, 61312, 232792, 874864, 3260360, 12064928, 44378984, 162399504, 591613880, 2146724864, 7762397576, 27980907248, 100580448920, 360636908000, 1290131211432, 4605675085008, 16410645183928

Offset

0,3

Comments

a(n)=Sum(A181327(n,k), k>=0).

References

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.

Links

Table of n, a(n) for n=0..24.

Index entries for linear recurrences with constant coefficients, signature (6,-5,-16,8,8,-4).

Formula

G.f. = z^2*(1-z)^2*(3-z^2)/[(1+z)(1-4z+2z^2)]^2.

a(n) = (3*A181326(n-1) -A181326(n-3))/2. - R. J. Mathar, Jul 24 2022

Example

a(2)=3 because in (0/2), (1/1), (2,0), (1,0 / 0,1), (0,1 / 1,0), (1,1 / 0,0), and (0,0 / 1,1) (the 2-compositions are written as (top row / bottom row)) we have 1+1+1+0+0+0+0=3 columns with even sums.

Maple

g := z^2*(1-z)^2*(3-z^2)/((1+z)^2*(1-4*z+2*z^2)^2): gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 27);

Prog

(PARI) a(n)=if(n, ([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; -4, 8, 8, -16, -5, 6]^(n-1)*[0; 3; 12; 59; 248; 1024])[1, 1], 0) \\ Charles R Greathouse IV, May 30 2026

Crossrefs

Cf. A181327, A181326.

Sequence in context: A298419 A385950 A126959 * A058861 A105668 A378574

Adjacent sequences: A181325 A181326 A181327 * A181329 A181330 A181331

Keyword

nonn,easy

Author

Emeric Deutsch, Oct 13 2010

Status

approved