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A052536
Number of compositions of n when parts 1 and 2 are of two kinds.
10
1, 2, 6, 17, 49, 141, 406, 1169, 3366, 9692, 27907, 80355, 231373, 666212, 1918281, 5523470, 15904198, 45794313, 131859469, 379674209, 1093228314, 3147825473, 9063802210, 26098178316, 75146709475, 216376326215, 623030800329
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
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
Row sums of A105478.
Sequence in context: A036365 A299162 A244400 * A122100 A122099 A026165
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