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

0,2

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

Row sums of A105478.

Cf. A105478, A076264.

Sequence in context: A036365 A299162 A244400 * A122100 A122099 A026165

Adjacent sequences: A052533 A052534 A052535 * A052537 A052538 A052539

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