A304159 a(n) = 2*n^3 - 4*n^2 + 6*n - 2 (n>=1).

2, 10, 34, 86, 178, 322, 530, 814, 1186, 1658, 2242, 2950, 3794, 4786, 5938, 7262, 8770, 10474, 12386, 14518, 16882, 19490, 22354, 25486, 28898, 32602, 36610, 40934, 45586, 50578, 55922, 61630, 67714, 74186, 81058, 88342, 96050, 104194, 112786, 121838, 131362, 141370, 151874, 162886, 174418, 186482, 199090

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)

a(n) = A033431(n) - A002943(n-1) = A033431(n) - 2*A014105(n-1). - Omar E. Pol, May 09 2018

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

Cf. A002943, A014105, A033431, A100705, A304160.

Sequence in context: A316172 A108100 A059924 * A211905 A022498 A036799

Adjacent sequences: A304156 A304157 A304158 * A304160 A304161 A304162

Keyword

nonn,easy

Author

Emeric Deutsch, May 09 2018

Status

approved