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 1-4 of 4 results.

A338109 a(n)/A002939(n+1) is the Kirchhoff index of the join of the disjoint union of two complete graphs on n vertices with the empty graph on n+1 vertices.

Original entry on oeis.org

1, 60, 289, 796, 1689, 3076, 5065, 7764, 11281, 15724, 21201, 27820, 35689, 44916, 55609, 67876, 81825, 97564, 115201, 134844, 156601, 180580, 206889, 235636, 266929, 300876, 337585, 377164, 419721, 465364, 514201, 566340, 621889, 680956, 743649, 810076, 880345
Offset: 0

Views

Author

Rigoberto Florez, Oct 10 2020

Keywords

Comments

Equivalently, the graph can be described as the graph on 3*n + 1 vertices with labels 0..3*n and with i and j adjacent iff A011655(i + j) = 1.
These graphs are cographs.
The initial term a(0) = 1 has been included to agree with the formula. For the graph, it should be 0.

Examples

			The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
  [0, 1, 1, 0, 1, 1, 0]
  [1, 0, 0, 1, 1, 0, 1]
  [1, 0, 0, 1, 0, 1, 1]
  [0, 1, 1, 0, 1, 1, 0]
  [1, 1, 0, 1, 0, 0, 1]
  [1, 0, 1, 1, 0, 0, 1]
  [0, 1, 1, 0, 1, 1, 0]
a(2) = 289 because the Kirchhoff index of the graph is 289/30 = 289/A002939(3).
The first few Kirchhoff indices (n >= 1) as reduced fractions are 5, 289/30, 199/14, 563/30, 769/33, 5065/182, 647/20, 11281/306, 3931/95, 7067/154, 6955/138, 35689/650.

Links

Crossrefs

Cf. A002939, A011655, A204437, A338104.

Programs

  • Mathematica
    Table[1+10n+31n^2+18n^3,{n,30}]
  • PARI
    a(n)=1+10*n+31*n^2+18*n^3 \\ Charles R Greathouse IV, Oct 18 2022

Formula

a(n) = 1 + 10*n + 31*n^2 + 18*n^3.
From Stefano Spezia, Oct 10 2020: (Start)
G.f.: (1 + 56*x + 55*x^2 - 4*x^3)/(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. (End)

A338527 Number of spanning trees in the join of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.

Original entry on oeis.org

24, 13500, 34420736, 239148450000, 3520397039081472, 94458953432730437824, 4179422085120000000000000, 283894102615246085842939590912, 28059580711858187192007680000000000, 3870669526565955444680027453177986243584
Offset: 1

Views

Author

Rigoberto Florez, Nov 07 2020

Keywords

Comments

Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff i+j> 0 mod 3.
These graphs are cographs.

Examples

			The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
 [0, 1, 0, 0, 0, 1, 1, 1]
 [1, 0, 0, 0, 0, 1, 1, 1]
 [0, 0, 0, 1, 1, 1, 1, 1]
 [0, 0, 1, 0, 1, 1, 1, 1]
 [0, 0, 1, 1, 0, 1, 1, 1]
 [1, 1, 1, 1, 1, 0, 0, 0]
 [1, 1, 1, 1, 1, 0, 0, 0]
 [1, 1, 1, 1, 1, 0, 0, 0]
a(2) = 13500 because the graph has 13500 spanning trees.

Links

Crossrefs

Cf. A338104, A338109.

Programs

  • Mathematica
    Table[(n + 1)*(2 n + 2)^n*(2 n + 1)^(2 n - 1), {n, 1, 10}]

Formula

a(n) = (n + 1)*(2 n + 2)^n*(2 n + 1)^(2 n - 1).

Extensions

Offset changed by Georg Fischer, Nov 03 2023

A338588 a(n)/A002939(n+1) is the Kirchhoff index of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.

Original entry on oeis.org

2, 77, 334, 881, 1826, 3277, 5342, 8129, 11746, 16301, 21902, 28657, 36674, 46061, 56926, 69377, 83522, 99469, 117326, 137201, 159202, 183437, 210014, 239041, 270626, 304877, 341902, 381809, 424706, 470701, 519902
Offset: 0

Views

Author

Rigoberto Florez, Nov 07 2020

Keywords

Comments

Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff i+j>0 mod 3.
These graphs are cographs.
The initial term a(0) = 2 has been included to agree with the formula. For the graph, is not defined.

Examples

			The adjacency matrix of the graph associated with n = 2 is:
  [0, 1, 0, 0, 0, 1, 1, 1]
  [1, 0, 0, 0, 0, 1, 1, 1]
  [0, 0, 0, 1, 1, 1, 1, 1]
  [0, 0, 1, 0, 1, 1, 1, 1]
  [0, 0, 1, 1, 0, 1, 1, 1]
  [1, 1, 1, 1, 1, 0, 0, 0]
  [1, 1, 1, 1, 1, 0, 0, 0]
  [1, 1, 1, 1, 1, 0, 0, 0].
a(2) = 334 because the Kirchhoff index of the graph is 334/30=334/A002939(3).
The first few Kirchhoff indices (n >= 1) as reduced fractions are 77/12, 167/15, 881/56, 913/45, 3277/132, 2671/91, 8129/240, 5873/153, 16301/380, 10951/231.

Links

Crossrefs

Cf. A338527, A338104, A338109.

Programs

  • Mathematica
    Table[(18n^3+37n^2+20n+2), {n,0,30}]

Formula

a(n) = 18*n^3 + 37*n^2 + 20*n + 2.
G.f.: (2 + 69*x + 38*x^2 - x^3)/(x - 1)^4.
E.g.f.: exp(x)*(2 + 75*x + 91*x^2 + 18*x^3). - Stefano Spezia, Nov 08 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Nov 08 2020

A338110 Number of spanning trees in the join of the disjoint union of two complete graphs each on n vertices with the empty graph on n vertices.

Original entry on oeis.org

1, 128, 139968, 536870912, 5000000000000, 92442129447518208, 2988151979474457198592, 154742504910672534362390528, 12044329605471552321957641846784, 1342177280000000000000000000000000000, 206097683218942123873399068932507659403264, 42281678783395138381516145098915043145456549888
Offset: 1

Views

Author

Rigoberto Florez, Oct 10 2020

Keywords

Comments

Equivalently, the graph can be described as the graph on 3*n vertices with labels 0..3*n-1 and with i and j adjacent iff A011655(i + j) = 1.
These graphs are cographs.

Examples

			The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
  [0, 1, 1, 0, 1, 1]
  [1, 0, 0, 1, 1, 0]
  [1, 0, 0, 1, 0, 1]
  [0, 1, 1, 0, 1, 1]
  [1, 1, 0, 1, 0, 0]
  [1, 0, 1, 1, 0, 0]
a(2) = 128 because the graph has 128 spanning trees.

Links

Crossrefs

Cf. A011655, A193131, A204437, A338104.

Programs

  • Mathematica
    Table[n (2 n)^(3 (n - 1)), {n, 1, 10}]

Formula

a(n) = n*(2*n)^(3*(n - 1)).
a(n) = A193131(n)/3.
Showing 1-4 of 4 results.

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