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A384127 a(n) is the number of integer quintuples (a,b,c,d,e) satisfying a system of linear inequalities and congruences specified in the comments.

%I A384127 #22 Jun 04 2025 11:52:40
%S A384127 1,25,226,1000,3126,7877,17151,33602,60751,103127,166378,257402,
%T A384127 384478,557377,787503,1088004,1473903,1962229,2572128,3325004,4244630,
%U A384127 5357279,6691855,8280004,10156255,12358131,14926280,17904606,21340380,25284381,29791007,34918406
%N A384127 a(n) is the number of integer quintuples (a,b,c,d,e) satisfying a system of linear inequalities and congruences specified in the comments.
%C A384127 The inequalities are
%C A384127   n + a + b + c + d + e >= 0,
%C A384127   121*n + 61*a + 13*b - 23*c - 47*d - 59*e >= 0,
%C A384127   121*n + 13*a - 47*b - 59*c - 23*d + 61*e >= 0,
%C A384127   121*n - 23*a - 59*b + 13*c + 61*d - 47*e >= 0,
%C A384127   121*n - 47*a - 23*b + 61*c - 59*d + 13*e >= 0,
%C A384127   121*n - 59*a + 61*b - 47*c + 13*d - 23*e >= 0,
%C A384127 The congruences are
%C A384127   n + a + b + c + d + e == 0 (mod 12),
%C A384127   121*n + 61*a + 13*b - 23*c - 47*d - 59*e == 0 (mod 11).
%H A384127 Ray Chandler, <a href="/A384127/b384127.txt">Table of n, a(n) for n = 0..1000</a>
%H A384127 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 A384127 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 A384127 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,2,-5,10,-10,5,-1).
%e A384127 For n=0, the sole solution is (a,b,c,d,e) = (0,0,0,0,0) so a(0) = 1.
%e A384127 For n=1, the a(1)=25 solutions are (-3,3,-1,0,0), (-2,0,2,-1,0), (-2,1,-1,2,-1), (-1,-2,2,1,-1), (-2,1,0,-1,1), (-1,-1,0,1,0), (0,-3,0,3,-1), (-1,-1,1,-2,2), (-1,0,-2,1,1), (0,-2,-1,0,2), (0,-1,-3,0,3), (-1,0,3,-3,0), (-1,1,0,0,-1), (0,-1,0,2,-2), (-1,2,-2,0,0), (0,-1,1,-1,0), (0,0,-1,-1,1), (1,-2,-1,1,0), (0,1,1,-2,-1), (1,-1,1,0,-2), (1,0,-1,0,-1), (2,-1,-2,-1,1), (1,2,-1,-1,-2), (2,0,0,-2,-1), (3,0,0,-1,-3).
%t A384127 a[n_] :=
%t A384127   Sum[Sum[Sum[
%t A384127      Sum[Sum[Boole[
%t A384127         PossibleZeroQ[
%t A384127          Mod[-b + 3 n + 4 q1 + 3 q2 + q3 + 2 q4, 5]]], {q4, 0,
%t A384127         Floor[5 n - 5 b - q1 - q2 - q3]}], {q3, 0,
%t A384127        Floor[5 n - 5 b - q1 - q2]}], {q2, 0,
%t A384127       Floor[5 n - 5 b - q1]}], {q1, 0, Floor[5 n - 5 b]}], {b, 0,
%t A384127     Floor[n]}];
%t A384127 Table[a[j], {j, 0, 50}]
%Y A384127 Cf. A370349, A384295.
%K A384127 nonn,easy
%O A384127 0,2
%A A384127 _Jeffery Opoku_, May 19 2025
%E A384127 More terms from _Jinyuan Wang_, May 26 2025