OFFSET
1,1
COMMENTS
For n>=2, a(n) is the first Zagreb index of the graph KK_n, defined as 2 copies of the complete graph K_n, with one vertex from one copy joined to two vertices of the other copy (see the Stevanovic et al. reference, p. 396).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of KK_n is M(KK_n; x,y) = (n-2)^2*x^{n-1}*y^{n-1}+2*(n-2)*x^{n-1}*y^n + (n-1)*x^{n-1}*y^{n+1} + x^n*y^n +2*x^n*y^{n+1}.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
D. Stevanovic, I. Stankovic, and M. Milosevic, More on the relation between energy and Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 61, 2009, 395-401.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
From Colin Barker, May 09 2018: (Start)
G.f.: 2*x*(3 - 3*x + 5*x^2 + 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)
MATHEMATICA
Table[2n^3-4n^2+10n-2 , {n, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {6, 18, 46, 102}, 50] (* Harvey P. Dale, Oct 17 2022 *)
PROG
(PARI) Vec(2*x*(3 - 3*x + 5*x^2 + x^3) / (1 - x)^4 + O(x^60)) \\ Colin Barker, May 09 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 09 2018
STATUS
approved