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A370349 a(n) is the number of integer triples (x,y,z) satisfying a system of linear inequalities and congruences specified in the comments.

%I A370349 #89 Jun 04 2025 13:14:38
%S A370349 1,6,18,39,72,120,185,270,378,511,672,864,1089,1350,1650,1991,2376,
%T A370349 2808,3289,3822,4410,5055,5760,6528,7361,8262,9234,10279,11400,12600,
%U A370349 13881,15246,16698,18239,19872,21600,23425,25350,27378,29511,31752,34104,36569,39150,41850,44671,47616,50688,53889,57222
%N A370349 a(n) is the number of integer triples (x,y,z) satisfying a system of linear inequalities and congruences specified in the comments.
%C A370349 The inequalities are
%C A370349   n + x + y + z >= 0,
%C A370349   49*n + 13*x - 11*y - 23*z >= 0,
%C A370349   49*n - 11*x - 23*y + 13*z >= 0,
%C A370349   49*n - 23*x + 13*y - 11*z >= 0,
%C A370349 The congruences are
%C A370349   n + x + y + z == 0 (mod 12),
%C A370349   49*n + 13*x - 11*y - 23*z == 0 (mod 7).
%H A370349 Ray Chandler, <a href="/A370349/b370349.txt">Table of n, a(n) for n = 0..1000</a>
%H A370349 T. Huber, N. Mayes, J. Opoku, and D. Ye, <a href="https://arxiv.org/abs/2403.15967">Ramanujan type congruences for quotients of Klein forms</a>, arXiv:2403.15967 [math.NT], 2024.
%H A370349 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 A370349 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F A370349 a(n) = floor((10 + 24*n + 18*n^2 + 4*n^3)/9).
%F A370349 a(n) = (A141530(n+2) - A102283(n))/9. - _Stefano Spezia_, Feb 17 2024
%e A370349 For n=0, the sole solution is (x,y,z) = (0,0,0) so a(0) = 1.
%e A370349 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 A370349 n = Range[0, 500, 2];
%t A370349 Floor[(10 + 24*n + 18*n^2 + 4*n^3)/9]
%o A370349 (Python)
%o A370349 def A370349(n): return ((n<<2)+10)*(n+1)**2//9 # _Chai Wah Wu_, Mar 11 2024
%Y A370349 Cf. A002620, A102283, A141530, A384127, A384295.
%K A370349 nonn,easy
%O A370349 0,2
%A A370349 _Jeffery Opoku_, Feb 16 2024