A025828 Expansion of 1/((1-x^3)*(1-x^4)*(1-x^6)).

1, 0, 0, 1, 1, 0, 2, 1, 1, 2, 2, 1, 4, 2, 2, 4, 4, 2, 6, 4, 4, 6, 6, 4, 9, 6, 6, 9, 9, 6, 12, 9, 9, 12, 12, 9, 16, 12, 12, 16, 16, 12, 20, 16, 16, 20, 20, 16, 25, 20, 20, 25, 25, 20, 30, 25, 25, 30, 30, 25, 36, 30, 30, 36, 36, 30, 42, 36, 36, 42, 42

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

Comments

Number of partitions of n into parts 3, 4, and 6. - Joerg Arndt, Aug 25 2013

Links

• Kevin Woods, The generalized Frobenius problem via restricted partition functions, arXiv:2011.00600 [math.CO], 2021. See Example 1.4 p. 2.
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,1,-1,0,-1,-1,0,0,1).

Programs

Mathematica

CoefficientList[Series[1/((1-x^3)(1-x^4)(1-x^6)),{x,0,90}],x] (* Harvey P. Dale, Jul 10 2019 *)

PARI

a(n) = (48*(1-n%3)*(n\3+1)+2*n^2+26*n+213+3*(2*n+13)*(-1)^n)\288;  \\ Tani Akinari, Aug 25 2013

PARI

Vec(1/((1-x^3)*(1-x^4)*(1-x^6))+ O(x^80)) \\ Michel Marcus, Nov 04 2014

PARI

a(n) = (n^2 + n*(21 + 3*(-1)^n - 8*(n%3)) + 144 - 12*(n%6))\144; \\ Hoang Xuan Thanh, Aug 31 2025

Formula

From Hoang Xuan Thanh, Aug 31 2025: (Start)

a(n) = floor(( n^2 + n*(21 + 3*(-1)^n - 8*(n mod 3)) )/144 + 1 - (n mod 6)/12).

a(n) = (floor((n+3)/3) - floor((n+1)/4)) * (floor((n+3)/3) + floor((n+1)/4) -floor((n+1)/2)). (End)

Extensions

More terms from Michel Marcus, Nov 04 2014