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.

1, 25, 226, 1000, 3126, 7877, 17151, 33602, 60751, 103127, 166378, 257402, 384478, 557377, 787503, 1088004, 1473903, 1962229, 2572128, 3325004, 4244630, 5357279, 6691855, 8280004, 10156255, 12358131, 14926280, 17904606, 21340380, 25284381, 29791007, 34918406

Offset: 0

Author

Jeffery Opoku, May 19 2025

Keywords

• nonn
• easy

Comments

The inequalities are

n + a + b + c + d + e >= 0,

121*n + 61*a + 13*b - 23*c - 47*d - 59*e >= 0,

121*n + 13*a - 47*b - 59*c - 23*d + 61*e >= 0,

121*n - 23*a - 59*b + 13*c + 61*d - 47*e >= 0,

121*n - 47*a - 23*b + 61*c - 59*d + 13*e >= 0,

121*n - 59*a + 61*b - 47*c + 13*d - 23*e >= 0,

The congruences are

n + a + b + c + d + e == 0 (mod 12),

121*n + 61*a + 13*b - 23*c - 47*d - 59*e == 0 (mod 11).

Examples

			For n=0, the sole solution is (a,b,c,d,e) = (0,0,0,0,0) so a(0) = 1.
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).
		

Links

• Ray Chandler, Table of n, a(n) for n = 0..1000
• T. Huber, N. Mayes, J. Opoku, and D. Ye, Ramanujan type congruences for quotients of Klein forms, arXiv:2403.15967 [math.NT], 2024.
• 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 (5,-10,10,-5,2,-5,10,-10,5,-1).

Crossrefs

Cf. A370349, A384295.

Programs

Mathematica

a[n_] :=
  Sum[Sum[Sum[
     Sum[Sum[Boole[
        PossibleZeroQ[
         Mod[-b + 3 n + 4 q1 + 3 q2 + q3 + 2 q4, 5]]], {q4, 0,
        Floor[5 n - 5 b - q1 - q2 - q3]}], {q3, 0,
       Floor[5 n - 5 b - q1 - q2]}], {q2, 0,
      Floor[5 n - 5 b - q1]}], {q1, 0, Floor[5 n - 5 b]}], {b, 0,
    Floor[n]}];
Table[a[j], {j, 0, 50}]

Extensions

More terms from Jinyuan Wang, May 26 2025