Voozh

A304162

a(n) = n^4 - 3*n^3 + 9*n^2 - 7*n + 5 (n>=1).

5, 19, 65, 185, 445, 935, 1769, 3085, 5045, 7835, 11665, 16769, 23405, 31855, 42425, 55445, 71269, 90275, 112865, 139465, 170525, 206519, 247945, 295325, 349205, 410155, 478769, 555665, 641485, 736895, 842585, 959269, 1087685, 1228595, 1382785

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For n>=2, a(n) is the second 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 second Zagreb index of a simple connected graph is the sum of the degree products 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

Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.

Dragan Stevanovic, Ivan Stankovic, and Marko 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 (5,-10,10,-5,1).

FORMULA

From Colin Barker, May 10 2018: (Start)

G.f.: x*(5 - 6*x + 20*x^2 + 5*x^4)/(1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5. (End)

E.g.f.: -5 + exp(x)*(5 + 7*x^2 + 3*x^3 + x^4). - Elmo R. Oliveira, Sep 10 2025

MAPLE

seq(n^4-3*n^3+9*n^2-7*n+5, n = 1 .. 40);

MATHEMATICA

Table[n (n - 1) (n^2 - 2 n + 7) + 5, {n, 1, 40}] (* Bruno Berselli, May 10 2018 *)

PROG

(PARI) Vec(x*(5 - 6*x + 20*x^2 + 5*x^4) / (1 - x)^5 + O(x^60)) \\ Colin Barker, May 10 2018