A181367 Number of 2-compositions of n containing at least one 0 entry. 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.
2, 6, 22, 78, 272, 940, 3232, 11080, 37920, 129648, 443008, 1513248, 5168000, 17647552, 60258304, 205746304, 702484992, 2398480128, 8189016064, 27959235072, 95459170304, 325918735360, 1112757649408, 3799195224064
Offset: 1
Examples
a(2)=6 because the 2-compositions of 2, written as (top row / bottom row), are (1/1), (0/2), (2/0), (1,0/0,1), (0,1/1,0), (1,1/0,0), (0,0/1,1) and only the first one does not contain a 0 entry.
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
- Index entries for linear recurrences with constant coefficients, signature (6,-10,4).
Crossrefs
Cf. A181365
Programs
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Maple
G := 2*z*(1-z)^3/((1-2*z)*(1-4*z+2*z^2)): Gser := series(G, z = 0, 30): seq(coeff(Gser, z, n), n = 1 .. 25);
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Mathematica
CoefficientList[Series[(2x (1-x)^3)/((1-2x)(1-4x+2x^2)),{x,0,30}],x] (* Harvey P. Dale, Mar 29 2020 *)
Formula
G.f.=2z(1-z)^3/[(1-2z)(1-4z+2z^2)].
4*a(n) = 2*A007070(n)-2^n, n>1. - R. J. Mathar, Jul 22 2022
Comments