A384328 Expansion of 1 / ((1-x)^3 * (1-x^7)).

1, 3, 6, 10, 15, 21, 28, 37, 48, 61, 76, 93, 112, 133, 157, 184, 214, 247, 283, 322, 364, 410, 460, 514, 572, 634, 700, 770, 845, 925, 1010, 1100, 1195, 1295, 1400, 1511, 1628, 1751, 1880, 2015, 2156, 2303, 2457, 2618, 2786, 2961

Offset: 0

Author

Hoang Xuan Thanh, May 26 2025

Keywords

• nonn
• easy

Comments

Number of nonnegative integer solutions of equation x + y + z + 7*w=n.

a(n) is the number of partitions of n into parts 1 of three kinds and 7 (of one kind). - Joerg Arndt, May 28 2025

Links

• Hoang Xuan Thanh, Table of n, a(n) for n = 0..1999
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,1,-3,3,-1).

Crossrefs

Cf. A000292, A002623, A014125, A122046, A122047, A175724.

Programs

PARI

my(x='x+O('x^50)); Vec(1/((1-x)^3*(1-x^7))) \\ Michel Marcus, May 27 2025

Formula

a(n) = ((n+2) * (n+9) * (n+4) - (r+2) * (r-5) * (r-3)) / 42 where r = n mod 7.

a(n) = floor((n+3) * (n^2+12*n+26) / 42).