OFFSET
0,3
COMMENTS
Number of nonisomorphic n-element interval orders with no 3-element antichain. - Richard Stanley, Nov 21 2011
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 0; 1, 1, 1; 0, 1, 1] or of the 3 X 3 matrix [1, 0, 1; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the number of Motzkin n-paths of height <= 2. - Alois P. Heinz, Nov 24 2023
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2000
Michael D. Barrus, Weakly threshold graphs, arXiv preprint arXiv:1608.01358 [math.CO], 2016.
Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv:1105.3713 [math.CO], 2011.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).
FORMULA
a(n) = A024537(n-1), n>0. - R. J. Mathar, Jan 28 2010
a(n) = 3*a(n-1)-a(n-2)-a(n-3). G.f.: (1-2*x)/((1-x)*(1-2*x-x^2)). - Colin Barker, Apr 01 2012
a(n) = (2+(1-sqrt(2))^n+(1+sqrt(2))^n)/4. - Colin Barker, Mar 16 2016
MAPLE
read("transforms") :
L := [1, seq(2^i, i=0..30)] ;
AERATE(L, 1) ;
BINOMIAL(%) ; # R. J. Mathar, Sep 26 2011
MATHEMATICA
LinearRecurrence[{3, -1, -1}, {1, 1, 2}, 50] (* Jean-François Alcover, Feb 25 2017 *)
PROG
(PARI) Vec((1-2*x)/((1-x)*(1-2*x-x^2)) + O(x^50)) \\ Colin Barker, Mar 16 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Dec 19 2009
STATUS
approved