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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.
4
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
0,2
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.
LINKS
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
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)
EXAMPLE
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.
MATHEMATICA
Table[1+10n+31n^2+18n^3, {n, 30}]
PROG
(PARI) a(n)=1+10*n+31*n^2+18*n^3 \\ Charles R Greathouse IV, Oct 18 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rigoberto Florez, Oct 10 2020
STATUS
approved