A304161 a(n) = 2*n^3 - 4*n^2 + 10*n - 2 (n>=1).

6, 18, 46, 102, 198, 346, 558, 846, 1222, 1698, 2286, 2998, 3846, 4842, 5998, 7326, 8838, 10546, 12462, 14598, 16966, 19578, 22446, 25582, 28998, 32706, 36718, 41046, 45702, 50698, 56046, 61758, 67846, 74322, 81198, 88486, 96198, 104346, 112942, 121998

Offset: 1

Author

Emeric Deutsch, May 09 2018

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A033431, A054000.

Programs

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 *)

PARI

Vec(2*x*(3 - 3*x + 5*x^2 + x^3) / (1 - x)^4 + O(x^60)) \\ Colin Barker, May 09 2018

Formula

a(n) = A033431(n-1) + A054000(n+1). - Omar E. Pol, May 09 2018

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)