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.

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

0,1

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.

Links

Table of n, a(n) for n=0..30.

H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.

Eric Weisstein's World of Mathematics, Kirchhoff Index

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

Example

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.

Mathematica

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

Crossrefs

Cf. A338527, A338104, A338109.

Sequence in context: A041721 A048358 A124456 * A364696 A308373 A183578

Adjacent sequences: A338585 A338586 A338587 * A338589 A338590 A338591

Keyword

nonn,easy

Author

Rigoberto Florez, Nov 07 2020

Status

approved