OFFSET
1,1
COMMENTS
a(n) is the first Zagreb index of the Barbell graph B(n) (n>=3).
The Barbell graph B(n) is defined as two copies of the complete graph K(n) (n>=3), connected by a bridge.
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 the Barbell graph B(n) is M(B(n),x,y) = (n-1)(n-2)x^{n-1}*y^{n-1}+2(n-1)x^{n-1]*y^n + x^n*y^n.
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.
Eric Weisstein's World of Mathematics, Barbell Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 2 * A100705(n-1).
From Colin Barker, May 09 2018: (Start)
G.f.: 2*x*(1 + x + 3*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)
MAPLE
seq(2*n^3-4*n^2+6*n-2, n = 1 .. 40);
MATHEMATICA
Table[2n^3-4n^2+6n-2 , {n, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {2, 10, 34, 86}, 50] (* Harvey P. Dale, Mar 05 2023 *)
PROG
(PARI) Vec(2*x*(1 + x + 3*x^2 + x^3) / (1 - x)^4 + O(x^60)) \\ Colin Barker, May 09 2018
(PARI) a(n) = 2*n^3-4*n^2+6*n-2; \\ Altug Alkan, May 09 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 09 2018
STATUS
approved