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A384295 a(n) is the number of integer sextuples (a,b,c,d,e,f) satisfying a system of linear inequalities and congruences specified in the comments.

%I A384295 #22 Jun 04 2025 10:12:01
%S A384295 1,42,684,4388,17976,56076,145630,331410,682596,1300338,2326422,
%T A384295 3952896,6432777,10091748,15340947,22690710,32765418,46319334,
%U A384295 64253491,87633588,117708960,155932526,203981823,263781030,337524061,427698636,537111456,668914338,826631436
%N A384295 a(n) is the number of integer sextuples (a,b,c,d,e,f) satisfying a system of linear inequalities and congruences specified in the comments.
%C A384295 The inequalities are
%C A384295   n + a + b + c + d + e + f >= 0,
%C A384295   169*n + 97*a + 37*b - 11*c - 47*d - 71*e - 83*f >= 0,
%C A384295   169*n + 37*a - 47*b - 83*c - 71*d - 11*e + 97*f >= 0,
%C A384295   169*n - 11*a - 83*b - 47*c + 97*d + 37*e - 71*f >= 0,
%C A384295   169*n - 47*a - 71*b + 97*c - 11*d - 83*e + 37*f >= 0,
%C A384295   169*n - 71*a - 11*b + 37*c - 83*d + 97*e - 47*f >= 0,
%C A384295   169*n - 83*a + 97*b - 71*c + 37*d - 47*e - 11*f >= 0.
%C A384295 The congruences are
%C A384295   n + a + b + c + d + e + f == 0 (mod 12),
%C A384295   169*n + 97*a + 37*b - 11*c - 47*d - 71*e - 83*f == 0 (mod 13).
%H A384295 Ray Chandler, <a href="/A384295/b384295.txt">Table of n, a(n) for n = 0..30</a>
%H A384295 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 A384295 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).
%e A384295 For n=0, the sole solution is (a,b,c,d,e,f) = (0,0,0,0,0,0) so a(0) = 1.
%e A384295 For n=1, the a(1)=42 solutions are (-3, 3, -1, 0, 0, 0), (-2, 0, 2, -1, 0, 0), (-2, 1, -1, 2, -1, 0), (-2, 1, 0, -1, 2, -1), (-2, 1, 0, 0, -1, 1), (-1, -2, 2, 1, -1, 0), (-1, -1, 0, 1, 1, -1), (-1, -1, 1, -1, 1, 0), (-1, -1, 1, 0, -2, 2), (-1, 0, -2, 2, 0, 0), (-1, 0, -1, 0, 0, 1), (-1, 0, 0, -3, 3, 0), (-1, 0, 1, 1, -2, 0), (-1, 0, 2, -2, 1, -1), (-1, 1, -1, 1, 0, -1), (-1, 1, 0, -1, 0, 0), (-1, 2, -2, 0, -1, 1), (0, -3, 0, 3, 0, -1), (0, -2, -1, 1, 2, -1), (0, -2, 0, 0, -1, 2), (0, -1, -2, 0, 1, 1), (0, -1, -1, -2, 1, 2), (0, -1, 0, 0, 2, -2), (0, -1, 0, 1, -1, 0), (0, -1, 1, -1, -1, 1), (0, -1, 3, 0, -3, 0), (0, 0, -3, -1, 0, 3), (0, 0, -1, -1, 1, 0), (0, 0, 1, 0, -1, -1), (0, 1, 0, -2, 1, -1), (0, 2, 0, 0, -2, -1), (1, -2, -1, 1, 0, 0), (1, -1, -1, 2, 0, -2), (1, -1, 0, 0, 0, -1), (1, 0, -1, -1, -1, 1), (1, 0, 1, -1, 0, -2), (1, 1, -1, 0, -1, -1), (1, 2, 0, -1, -1, -2), (2, -1, -2, -1, 0, 1), (2, 0, -1, -2, 0, 0), (2, 0, 1, -1, -2, -1), (3, 0, 0, 0, -1, -3).
%t A384295 a[n_]:=Sum[Boole[Mod[12*n+25*b-11*t1+9*t2-7*t3+2*t4-6*t5,19]==0],{b,0,Floor[7*n/6]},{t1,0,Floor[7*n-6*b]},{t2,0,Floor[7*n-6*b-t1]},{t3,0,Floor[7*n-6*b-t1-t2]},{t4,0,Floor[7*n-6*b-t1-t2-t3]},{t5,0,Floor[7*n-6*b-t1-t2-t3-t4]}];
%t A384295 Table[a[j],{j,0,20}]
%Y A384295 Cf. A370349, A384127.
%K A384295 nonn
%O A384295 0,2
%A A384295 _Jeffery Opoku_, May 24 2025
%E A384295 More terms from _Jinyuan Wang_, May 26 2025
%E A384295 a(29) and a(30) from _Ray Chandler_, Jun 04 2025