cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A288961 Number of 3-cycles in the n X n rook graph.

Original entry on oeis.org

0, 0, 6, 32, 100, 240, 490, 896, 1512, 2400, 3630, 5280, 7436, 10192, 13650, 17920, 23120, 29376, 36822, 45600, 55860, 67760, 81466, 97152, 115000, 135200, 157950, 183456, 211932, 243600, 278690, 317440, 360096, 406912, 458150, 514080, 574980, 641136, 712842, 790400
Offset: 1

Views

Author

Eric W. Weisstein, Jun 20 2017

Keywords

Links

Crossrefs

Cf. A288962 (4-cycles), A288963 (5-cycles), A288960 (6-cycles).
Main diagonal of A360855.

Programs

  • Mathematica
    Table[n^2 (n - 1) (n - 2)/3, {n, 20}]
Table[2 n Binomial[n, 3], {n, 20}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 6, 32, 100}, 20]
CoefficientList[Series[-((2 x^2 (3 + x))/(-1 + x)^5), {x, 0, 20}], x]
  • PARI
    a(n) = {2*n*binomial(n,3)} \\ Andrew Howroyd, Apr 26 2020

Formula

a(n) = 2*n*binomial(n,3).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).
G.f.: (-2*x^3*(3+x))/(-1+x)^5.

Extensions

Terms a(31) and beyond from Andrew Howroyd, Apr 26 2020

A288960 Number of 6-cycles in the n X n rook graph.

Original entry on oeis.org

0, 0, 60, 1248, 8400, 35520, 114660, 309120, 731808, 1569600, 3114540, 5802720, 10261680, 17367168, 28310100, 44674560, 68527680, 102522240, 150012828, 215186400, 303208080, 420383040, 574335300, 774204288, 1030860000, 1357137600, 1768092300, 2281275360, 2917032048, 3698822400
Offset: 1

Views

Author

Eric W. Weisstein, Jun 20 2017

Keywords

Links

Crossrefs

Cf. A288961 (3-cycles), A288962 (4-cycles), A288963 (5-cycles).

Programs

  • Mathematica
    Table[(n - 1) (n - 2) n^2 (n + 2) (n^2 + 2 n - 11)/6, {n, 20}]
Table[Binomial[n, 3] n (n + 2) (n^2 + 2 n - 11), {n, 20}]
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 60, 1248, 8400, 35520, 114660, 309120}, 20]
CoefficientList[Series[(12 x^2 (5 + 64 x + 8 x^2 - 8 x^3 + x^4))/(-1 + x)^8, {x, 0, 20}], x]

Formula

a(n) = (n-1)*(n-2)*n^2*(n+2)*(n^2+2*n-11)/6.
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)+a(n-8).
G.f.: (12*x^3*(5+64*x+8*x^2-8*x^3+x^4))/(-1+x)^8.

A288962 Number of 4-cycles in the n X n rook graph.

Original entry on oeis.org

0, 1, 9, 60, 250, 765, 1911, 4144, 8100, 14625, 24805, 39996, 61854, 92365, 133875, 189120, 261256, 353889, 471105, 617500, 798210, 1018941, 1285999, 1606320, 1987500, 2437825, 2966301, 3582684, 4297510, 5122125, 6068715, 7150336, 8380944, 9775425, 11349625, 13120380, 15105546, 17324029, 19795815, 22542000
Offset: 1

Views

Author

Eric W. Weisstein, Jun 20 2017

Keywords

Links

Crossrefs

Cf. A288961 (3-cycles), A288963 (5-cycles), A288960 (6-cycles).

Programs

  • Magma
    [n^2*(n-1)*(n^2-4*n+5)/4 : n in [1..50]]; // Wesley Ivan Hurt, Apr 23 2021
  • Mathematica
    Table[n^2 (n - 1) (n^2 - 4 n + 5)/4, {n, 20}]
Table[n Binomial[n, 2] (n^2 - 4 n + 5)/2, {n, 20}]
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 9, 60, 250, 765}, 20]
CoefficientList[Series[(x (1 + 3 x + 21 x^2 + 5 x^3))/(-1 + x)^6, {x, 0, 20}], x]

Formula

a(n) = n*binomial(n,2)*(n^2-4*n+5)/2.
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6).
G.f.: (x^2*(1+3*x+21*x^2+5*x^3))/(-1+x)^6.
Showing 1-3 of 3 results.

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