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a(n) = 591*2^n - 273.
1

%I #15 Sep 08 2022 08:46:18

%S 318,909,2091,4455,9183,18639,37551,75375,151023,302319,604911,

%T 1210095,2420463,4841199,9682671,19365615,38731503,77463279,154926831,

%U 309853935,619708143,1239416559,2478833391,4957667055,9915334383,19830669039,39661338351

%N a(n) = 591*2^n - 273.

%C a(n) is the second Zagreb index of the phenylazomethine dendrimer G[n] defined pictorially in the Yarahmadi references. 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.

%C The M-polynomial of the dendrimer NSB[n] is M(NSB[n], x, y) = 9*2^n*x*y^4 + (24*2^n - 12)*x^2*y^2 + (48*2^n -24)*x^2*y^3 +(15*2^n-9)*x^3*y^3+3*2^n*x^3*y^4.

%H E. Deutsch and Sandi Klavzar, <a href="http://dx.doi.org/10.22052/ijmc.2015.10106">M-polynomial and degree-based topological indices</a>, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.

%H Z. Yarahmadi, <a href="http://dx.doi.org/10.22052/ijmc.2010.5154">Eccentric connectivity and augmented eccentric connectivity indices of N-branches phenylacetylenes nanostar dendrimers</a>, Iranian J. Math. Chem., 1, No. 2, 2010, 105-110.

%H Z. Yarahmadi and G. H. Fath-Tabar, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match65/n1/match65n1_201-208.pdf">The Wiener, Szeged, PI, Vertex PI, the first and second Zagreb indices of N-branched phenylacetylenes dendrimers</a>, MATCH: Commun. Math. Comput. Chem, 65 (2011) 201-208.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).

%F O.g.f.: 3*(106 - 15*x)/((1 - x)*(1 - 2*x)).

%F E.g.f.: 3*(-91 + 197*exp(x))*exp(x).

%F a(n) = 3*a(n-1) - 2*a(n-2).

%p seq(591*2^n-273, n=0..35);

%t Table[591 2^n - 273, {n, 0, 35}] (* _Vincenzo Librandi_, Nov 16 2016 *)

%o (Magma) [591*2^n-273: n in [0..35]]; // _Vincenzo Librandi_, Nov 16 2016

%Y Cf. A278130.

%K nonn,easy

%O 0,1

%A _Emeric Deutsch_, Nov 15 2016

%E Edited by _Bruno Berselli_, Nov 16 2016