A370349 -id:A370349 - cp's OEIS frontend

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

Jeffery Opoku, May 19 2025

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).

			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).

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).

Cf. A370349, A384295.

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}]

More terms from Jinyuan Wang, May 26 2025

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.

Original entry on oeis.org

1, 42, 684, 4388, 17976, 56076, 145630, 331410, 682596, 1300338, 2326422, 3952896, 6432777, 10091748, 15340947, 22690710, 32765418, 46319334, 64253491, 87633588, 117708960, 155932526, 203981823, 263781030, 337524061, 427698636, 537111456, 668914338, 826631436

Offset: 0

Jeffery Opoku, May 24 2025

nonn

The inequalities are
  n + a + b + c + d + e + f >= 0,
  169*n + 97*a + 37*b - 11*c - 47*d - 71*e - 83*f >= 0,
  169*n + 37*a - 47*b - 83*c - 71*d - 11*e + 97*f >= 0,
  169*n - 11*a - 83*b - 47*c + 97*d + 37*e - 71*f >= 0,
  169*n - 47*a - 71*b + 97*c - 11*d - 83*e + 37*f >= 0,
  169*n - 71*a - 11*b + 37*c - 83*d + 97*e - 47*f >= 0,
  169*n - 83*a + 97*b - 71*c + 37*d - 47*e - 11*f >= 0.
The congruences are
  n + a + b + c + d + e + f == 0 (mod 12),
  169*n + 97*a + 37*b - 11*c - 47*d - 71*e - 83*f == 0 (mod 13).

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

Ray Chandler, Table of n, a(n) for n = 0..30
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).

Cf. A370349, A384127.

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]}];
Table[a[j],{j,0,20}]

More terms from Jinyuan Wang, May 26 2025

a(29) and a(30) from Ray Chandler, Jun 04 2025

cp's OEIS Frontend

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.

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