A305272 a(n) = 836*2^n - 676.

160, 996, 2668, 6012, 12700, 26076, 52828, 106332, 213340, 427356, 855388, 1711452, 3423580, 6847836, 13696348, 27393372, 54787420, 109575516, 219151708, 438304092, 876608860, 1753218396, 3506437468, 7012875612, 14025751900, 28051504476, 56103009628, 112206019932, 224412040540, 448824081756

Offset: 0

Author

Emeric Deutsch, May 30 2018

Keywords

• nonn
• easy

Comments

a(n) is the second Zagreb index of the polyphenylene dendrimer G[n], defined pictorially in the Arif et al. reference (see Fig. 1, where G[2] is shown).

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 the polyphenylene dendrimer G[n] is M(G[n]; x, y) = (56*2^n - 40)*x^2*y^2 + (48*2^n - 40)*x^2*y^3 +(36* 2^n - 36)*x^3*y^3 + 4*x^3 *y^4.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• N. E. Arif, Roslan Hasni and Saeid Alikhani, Fourth order and fourth sum connectivity indices of polyphenylene dendrimers, J. Applied Science, 12 (21), 2012, 2279-2282.
• E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A305269, A305270, A305271.

Programs

Maple

seq(836*2^n-676, n = 0..40);

Mathematica

836*2^Range[0,40]-676 (* or  *) LinearRecurrence[{3,-2},{160,996},40] (* Harvey P. Dale, Jun 19 2021 *)

PARI

Vec(4*(40 + 129*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 31 2018

Formula

From Colin Barker, May 31 2018: (Start)

G.f.: 4*(40 + 129*x) / ((1 - x)*(1 - 2*x)).

a(n) = 3*a(n-1) - 2*a(n-2) for n>1.

(End)