A384127 -id:A384127 - cp's OEIS frontend

A370349 a(n) is the number of integer triples (x,y,z) satisfying a system of linear inequalities and congruences specified in the comments.

1, 6, 18, 39, 72, 120, 185, 270, 378, 511, 672, 864, 1089, 1350, 1650, 1991, 2376, 2808, 3289, 3822, 4410, 5055, 5760, 6528, 7361, 8262, 9234, 10279, 11400, 12600, 13881, 15246, 16698, 18239, 19872, 21600, 23425, 25350, 27378, 29511, 31752, 34104, 36569, 39150, 41850, 44671, 47616, 50688, 53889, 57222

Offset: 0

Jeffery Opoku, Feb 16 2024

The inequalities are
  n + x + y + z >= 0,
  49*n + 13*x - 11*y - 23*z >= 0,
  49*n - 11*x - 23*y + 13*z >= 0,
  49*n - 23*x + 13*y - 11*z >= 0,
The congruences are
  n + x + y + z == 0 (mod 12),
  49*n + 13*x - 11*y - 23*z == 0 (mod 7).

			For n=0, the sole solution is (x,y,z) = (0,0,0) so a(0) = 1.
For n=1, the a(1)=6 solutions are (-1, -3, 3), (-2, 0, 1), (-3, 3, -1), (1, -2, 0), (0, 1, -2), (3, -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 (3,-3,2,-3,3,-1).

Cf. A002620, A102283, A141530, A384127, A384295.

n = Range[0, 500, 2];
Floor[(10 + 24*n + 18*n^2 + 4*n^3)/9]

def A370349(n): return ((n<<2)+10)*(n+1)**2//9 # Chai Wah Wu, Mar 11 2024

a(n) = floor((10 + 24*n + 18*n^2 + 4*n^3)/9).

a(n) = (A141530(n+2) - A102283(n))/9. - Stefano Spezia, Feb 17 2024

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

A370349 a(n) is the number of integer triples (x,y,z) satisfying a system of linear inequalities and congruences specified in the comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula