A052536 Number of compositions of n when parts 1 and 2 are of two kinds.

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

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

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

Examples

			a(2)=6 because we have (2),(2'),(1,1),(1,1'),(1',1) and (1',1').
		

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).

Crossrefs

Row sums of A105478.

Cf. A105478, A076264.

Programs

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 *)

PARI

Vec((1-x)/(1-3*x+x^3)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 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

Extensions

More terms from James Sellers, Jun 06 2000

Edited by Emeric Deutsch, Apr 10 2005

More terms from Gary W. Adamson, Sep 05 2005