OFFSET
0,2
COMMENTS
The g.f. for compositions of k_1 kinds of 1's, k_2 kinds of 2's, ..., k_j kinds of j's, ... is 1/(1 - Sum_{j>=1} k_j * x^j). - Joerg Arndt, Jul 06 2011
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..2177
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 467
Index entries for linear recurrences with constant coefficients, signature (3,0,-1).
FORMULA
G.f.: (1-x)/(1 - 3*x + x^3).
G.f.: 1/(1 - (2*x + 2*x^2 + Sum_{j>=3} x^j)). - Joerg Arndt, Jul 06 2011
a(n) = Sum(-(1/9)*(-2 + r^2 - r)*r^(-1-n)), r = RootOf(1 - 3*x + x^3).
a(0)=1, a(1)=2, a(2)=6, a(n) = 3*a(n-1) - a(n-3) for n >= 3. - Emeric Deutsch, Apr 10 2005
a(n) = left term in M^n * [1 0 0], where M = the 3 X 3 matrix [2 1 1 / 1 1 0 / 1 0 0]. Right term in M^n *[1 0 0] is a(n-1); middle term is A076264(n-1). - Gary W. Adamson, Sep 05 2005
3*a(n) = A123891(n+1). - Jeffrey R. Goodwin, Jul 03 2011
EXAMPLE
a(2)=6 because we have (2),(2'),(1,1),(1,1'),(1',1) and (1',1').
MAPLE
spec := [S, {S=Sequence(Union(Z, Prod(Z, Union(Z, Sequence(Z)))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
a[0] = 1; a[1] = 2; a[2] = 6; a[n_] := a[n] = 3*a[n-1] - a[n-3]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jun 12 2013, after Emeric Deutsch *)
PROG
(PARI) Vec((1-x)/(1-3*x+x^3)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 06 2000
Edited by Emeric Deutsch, Apr 10 2005
More terms from Gary W. Adamson, Sep 05 2005
STATUS
approved