A370349 - OEIS

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).

%I #82 Mar 12 2024 02:33:05

%S 1,6,18,39,72,120,185,270,378,511,672,864,1089,1350,1650,1991,2376,

%T 2808,3289,3822,4410,5055,5760,6528,7361,8262,9234,10279,11400,12600,

%U 13881,15246,16698,18239,19872,21600,23425,25350,27378,29511,31752,34104,36569,39150,41850,44671,47616,50688,53889,57222

%N 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).

%H T. Huber, N. Mayes, J. Opoku, and D. Ye, <a href="https://doi.org/10.1016/j.jnt.2023.11.009">Ramanujan type congruences for quotients of Klein forms</a>, Journal of Number Theory, 258, 281-333, (2024).

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).

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

%F a(n) = (A141530(n+2) - A102283(n))/9. - _Stefano Spezia_, Feb 17 2024

%e For n=0, the sole solution is (x,y,z) = (0,0,0) so a(0) = 1.

%e 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).

%t n = Range[0, 500, 2];

%t Floor[(10 + 24*n + 18*n^2 + 4*n^3)/9]

%o (Python)

%o def A370349(n): return ((n<<2)+10)*(n+1)**2//9 # _Chai Wah Wu_, Mar 11 2024

%Y Cf. A002620, A102283, A141530.

%K nonn,easy

%O 0,2

%A _Jeffery Opoku_, Feb 16 2024