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 1-5 of 5 results.

A360855 Array read by antidiagonals: T(m,n) is the number of triangles in the rook graph K_m X K_n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 4, 2, 2, 4, 10, 8, 6, 8, 10, 20, 20, 16, 16, 20, 20, 35, 40, 35, 32, 35, 40, 35, 56, 70, 66, 60, 60, 66, 70, 56, 84, 112, 112, 104, 100, 104, 112, 112, 84, 120, 168, 176, 168, 160, 160, 168, 176, 168, 120, 165, 240, 261, 256, 245, 240, 245, 256, 261, 240, 165
Offset: 1

Views

Author

Andrew Howroyd, Feb 24 2023

Keywords

Comments

A triangle is a clique of size 3. Also, a 3-cycle.

Examples

			Array begins:
=======================================
m\n|  1   2   3   4   5   6   7   8 ...
---+-----------------------------------
1  |  0   0   1   4  10  20  35  56 ...
2  |  0   0   2   8  20  40  70 112 ...
3  |  1   2   6  16  35  66 112 176 ...
4  |  4   8  16  32  60 104 168 256 ...
5  | 10  20  35  60 100 160 245 360 ...
6  | 20  40  66 104 160 240 350 496 ...
7  | 35  70 112 168 245 350 490 672 ...
8  | 56 112 176 256 360 496 672 896 ...
  ...

Links

Crossrefs

Main diagonal is A288961.
Rows n=1..3 are A000292(n-2), A007290, A060354.
Cf. A286418, A360853.

Programs

  • PARI
    T(m, n) = m*binomial(n,3) + n*binomial(m,3)

Formula

T(m,n) = m*binomial(n,3) + n*binomial(m,3).
T(m,n) = T(n,m).

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.

A288963 Number of 5-cycles in the n X n rook graph.

Original entry on oeis.org

0, 0, 36, 288, 1320, 4464, 12348, 29568, 63504, 125280, 230868, 402336, 669240, 1070160, 1654380, 2483712, 3634464, 5199552, 7290756, 10041120, 13607496, 18173232, 23951004, 31185792, 40158000, 51186720, 64633140, 80904096, 100455768, 123797520
Offset: 1

Views

Author

Eric W. Weisstein, Jun 20 2017

Keywords

Links

Crossrefs

Cf. A288961 (3-cycles), A288962 (4-cycles), A288960 (6-cycles).

Programs

  • Mathematica
    Table[(n - 2) (n - 1) n^2 (n^2 - 2 n + 7)/5, {n, 20}]
Table[6 n Binomial[n, 3] (n^2 - 2 n + 7)/5, {n, 20}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 36, 288, 1320, 4464, 12348}, 20]
CoefficientList[Series[-((12 x^2 (3 + 3 x + 5 x^2 + x^3))/(-1 + x)^7), {x, 0, 20}], x]

Formula

a(n) = 6*n*binomial(n,3)*(n^2-2*n+7)/5.
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).
G.f.: (-12*x^3*(3+3*x+5*x^2+x^3))/(-1+x)^7.

A360854 Number of induced cycles in the n X n rook graph.

Original entry on oeis.org

0, 1, 21, 236, 4040, 114105, 4662721, 256485936, 18226110456, 1623855703785, 177195820502965, 23237493232958796, 3605437233380103056, 653193551573628910481, 136634950180317224879985, 32681589590709963123110080, 8863149183726257535369656976
Offset: 1

Views

Author

Andrew Howroyd, Feb 24 2023

Keywords

Comments

Induced cycles are sometimes called chordless cycles (but some definitions require chordless cycles to have a cycle length of at least 4). See A070968 for the version that excludes triangles.

Links

Crossrefs

Main diagonal of A360853.
Cf. A070968, A234624, A288961, A360852.

Programs

  • PARI
    a(n) = 2*n*binomial(n,3) + sum(k=2, n, binomial(n,k)^2 * k! * (k-1)!)/2

Formula

a(n) = A288961(n) + A070968(n).
a(n) = 2*n*binomial(n,3) + Sum_{k=2..n} binomial(n,k)^2 * k! * (k-1)! / 2.
Showing 1-5 of 5 results.

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