A025796 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^6)).

1, 0, 1, 1, 1, 1, 3, 1, 3, 3, 3, 3, 6, 3, 6, 6, 6, 6, 10, 6, 10, 10, 10, 10, 15, 10, 15, 15, 15, 15, 21, 15, 21, 21, 21, 21, 28, 21, 28, 28, 28, 28, 36, 28, 36, 36, 36, 36, 45, 36, 45, 45, 45, 45, 55, 45, 55, 55, 55, 55, 66, 55, 66, 66, 66, 66, 78, 66, 78, 78, 78

Offset

0,7

Comments

Number of partitions of n into parts 2, 3, and 6. - Hoang Xuan Thanh, Aug 21 2025

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1,1,0,-1,-1,0,1).

Formula

a(6n) = a(6n+2) = a(6n+3) = a(6n+4) = a(6n+5) = n*(n+1)/2. a(6n+1) = (n-1)*n/2. - Franklin T. Adams-Watters, Oct 27 2014

a(n) = binomial(floor(((floor(n/2) - (n mod 2))/3)) + 2, 2). - Hoang Xuan Thanh, Aug 25 2025

From Amiram Eldar, Oct 12 2025: (Start)

Sum_{n>=2} 1/a(n) = 11.

Sum_{n>=2} (-1)^(n+1)/a(n) = 1. (End)

Mathematica

A025796[n_] := Binomial[Quotient[Quotient[n, 2] - Mod[n, 2], 3] + 2, 2];
Array[A025796, 100, 0] (* Paolo Xausa, Sep 28 2025 *)
CoefficientList[Series[1/((1-x^2)(1-x^3)(1-x^6)), {x, 0, 80}], x] (* or  *) LinearRecurrence[{0, 1, 1, 0, -1, 1, 0, -1, -1, 0, 1}, {1, 0, 1, 1, 1, 1, 3, 1, 3, 3, 3}, 80] (* Harvey P. Dale, May 11 2026 *)

Prog

(PARI) a(n)=(n^2+(4*((n+2)%3)+7+3*(-1)^n)*n+57+33*(-1)^n/2)\72 \\ Tani Akinari, Oct 27 2014
(PARI) a(n) = binomial((n\2-n%2)\3+2, 2) \\ Hoang Xuan Thanh, Aug 21 2025

Crossrefs

Cf. A024163.

Sequence in context: A068074 A063195 A334070 * A024163 A029155 A042951

Adjacent sequences: A025793 A025794 A025795 * A025797 A025798 A025799

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved