A370349 a(n) is the number of integer triples (x,y,z) satisfying: n + x + y + z >= 0, 49*n + 13*x - 11*y - 23*z >= 0, 49*n - 11*x - 23*y + 13*z >= 0, 49*n - 23*x + 13*y - 11*z >= 0, n + x + y + z == 0 (mod 12), 49*n + 13*x - 11*y - 23*z == 0 (mod 7).

1, 6, 18, 39, 72, 120, 185, 270, 378, 511, 672, 864, 1089, 1350, 1650, 1991, 2376, 2808, 3289, 3822, 4410, 5055, 5760, 6528, 7361, 8262, 9234, 10279, 11400, 12600, 13881, 15246, 16698, 18239, 19872, 21600, 23425, 25350, 27378, 29511, 31752, 34104, 36569, 39150, 41850, 44671, 47616, 50688, 53889, 57222

Offset

0,2

Links

Table of n, a(n) for n=0..49.

T. Huber, N. Mayes, J. Opoku, and D. Ye, Ramanujan type congruences for quotients of Klein forms, Journal of Number Theory, 258, 281-333, (2024).

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Formula

a(n) = floor((10 + 24*n + 18*n^2 + 4*n^3)/9).

a(n) = (A141530(n+2) - A102283(n))/9. - Stefano Spezia, Feb 17 2024

Example

For n=0, the sole solution is (x,y,z) = (0,0,0) so a(0) = 1.
For n=1, the a(1)=6 solutions are (-1, -3, 3), (-2, 0, 1), (-3, 3, -1), (1, -2, 0), (0, 1, -2), (3, -1, -3).

Mathematica

n = Range[0, 500, 2];
Floor[(10 + 24*n + 18*n^2 + 4*n^3)/9]

Prog

(Python)
def A370349(n): return ((n<<2)+10)*(n+1)**2//9 # Chai Wah Wu, Mar 11 2024

Crossrefs

Cf. A002620, A102283, A141530.

Sequence in context: A129863 A272014 A272453 * A271252 A271541 A035489

Adjacent sequences: A370346 A370347 A370348 * A370350 A370351 A370352

Keyword

nonn,easy

Author

Jeffery Opoku, Feb 16 2024

Status

approved