OFFSET
0,2
COMMENTS
Number of compositions of n into two sorts of parts 1 and 3, and one sort of parts 2. - Joerg Arndt, May 02 2015
From F. Chapoton, Nov 05 2025: (Start)
For n odd, a(n) is even. For n even, a(n) is odd.
This sequence can be seen as the second term of an infinite sequence of sequences, the first one being A000129. Consecutive sequences are defined by longer and longer recurrences. (End)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..2354
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16; also arXiv preprint, arXiv:1302.2274 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (2,1,2).
FORMULA
a(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j) * binomial(k,j) * 2^(-n+4*k-2*j), n>0, a(0) = 1. - Vladimir Kruchinin, May 05 2011
a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3). - Stefan Schuster, Apr 24 2015
MATHEMATICA
LinearRecurrence[{2, 1, 2}, {1, 2, 5}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
CoefficientList[Series[1/(1-2x-x^2-2x^3), {x, 0, 30}], x] (* Harvey P. Dale, Jul 22 2024 *)
PROG
(Maxima) a(n):=sum(sum(binomial(j, n-3*k+2*j)*binomial(k, j)*2^(-n+4*k-2*j), j, 0, k), k, 1, n); /* Vladimir Kruchinin, May 05 2011 */
(PARI) Vec(1/(1-2*x-x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 17 2002
STATUS
approved