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
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
KEYWORD
nonn,easy
AUTHOR
Rigoberto Florez, Nov 07 2020
STATUS
approved