A384127 - OEIS

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

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OFFSET

0,2

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).

LINKS

Ray Chandler, Table of n, a(n) for n = 0..1000

Timothy Huber, Nathaniel Mayes, Jeffery Opoku, and Dongxi Ye, Ramanujan type congruences for quotients of Klein forms, arXiv:2403.15967 [math.NT], 2024.

Timothy Huber, Nathaniel Mayes, Jeffery Opoku, and Dongxi Ye, Ramanujan type congruences for quotients of Klein forms, Journal of Number Theory, Vol. 258 (2024), pp. 281-333.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,2,-5,10,-10,5,-1).

FORMULA

From Elmo R. Oliveira, May 13 2026: (Start)

G.f.: (1 + 20*x + 111*x^2 + 110*x^3 + 141*x^4 + 110*x^5 + 111*x^6 + 20*x^7 + x^8) / ((x - 1)^6 * (1 + x + x^2 + x^3 + x^4)).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 2*a(n-5) - 5*a(n-6) + 10*a(n-7) - 10*a(n-8) + 5*a(n-9) - a(n-10). (End)

EXAMPLE

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).

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}]

CROSSREFS