A176848 Number of compositions of n into floor(j/3) kinds of j's for all j>=1.

1, 0, 0, 1, 1, 1, 3, 4, 5, 10, 15, 21, 36, 56, 83, 134, 210, 320, 505, 791, 1221, 1911, 2988, 4639, 7240, 11305, 17595, 27436, 42806, 66691, 103968, 162144, 252720, 393965, 614285, 957581, 1492791, 2327396, 3628273, 5656274, 8818275, 13747425, 21431700, 33411976, 52088551, 81204526, 126596778, 197361904, 307682405

Offset: 0

Author

Joerg Arndt, Jul 06 2011

Keywords

• 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 )).

Links

• Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, Mark Shattuck, Some New Restricted n-Color Composition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 2, -1).

Crossrefs

Cf. A121907 (floor((3*j-1)/2)), A055841 (3*j-1), A052156 (2*j-1), A006053 (floor(j/2)), A143787 (floor((3*j)/2)).

Programs

PARI

N=66; x='x+O('x^N) /* that many terms */
gf= 1/(1-sum(j=1,N, floor(j/3)*x^j ))
Vec(gf) /* show terms */

Formula

G.f.: 1/(1-sum(j>=1, floor(j/3)*x^j )).

Conjectural g.f.: (x-1)^2*(x^2+x+1) / (x^4-2*x^3-x+1). - Colin Barker, May 15 2013

G.f.: 1 + x^3*Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+1 + 2*x^2 - x^3)/( x*(4*k+3 + 2*x^2 - x^3 ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 11 2013