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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Jeffery Opoku

Jeffery Opoku's wiki page.

Jeffery Opoku has authored 3 sequences.

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

Views

Author

Jeffery Opoku, May 24 2025

Keywords

Comments

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).

Examples

			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).

Links

Crossrefs

Cf. A370349, A384127.

Programs

  • Mathematica
    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}]

Extensions

More terms from Jinyuan Wang, May 26 2025
a(29) and a(30) from Ray Chandler, Jun 04 2025

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.

Original entry on oeis.org

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

Views

Author

Jeffery Opoku, May 19 2025

Keywords

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

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

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

Original entry on oeis.org

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

Views

Author

Jeffery Opoku, Feb 16 2024

Keywords

Comments

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).

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    n = Range[0, 500, 2];
Floor[(10 + 24*n + 18*n^2 + 4*n^3)/9]
  • Python
    def A370349(n): return ((n<<2)+10)*(n+1)**2//9 # Chai Wah Wu, Mar 11 2024

Formula

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

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