%I #14 Feb 12 2021 12:01:01
%S 132,812,2172,4892,10332,21212,42972,86492,173532,347612,695772,
%T 1392092,2784732,5570012,11140572,22281692,44563932,89128412,
%U 178257372,356515292,713031132,1426062812,2852126172,5704252892,11408506332,22817013212,45634026972,91268054492,182536109532,365072219612
%N a(n) = 680*2^n - 548.
%C a(n) is the first Zagreb index of the polyphenylene dendrimer G[n], defined pictorially in the Arif et al. reference (see Fig. 1, where G[2] is shown).
%C 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.
%C 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.
%H Colin Barker, <a href="/A305271/b305271.txt">Table of n, a(n) for n = 0..1000</a>
%H N. E. Arif, Roslan Hasni and Saeid Alikhani, <a href="http://dx.doi.org/10.3923/jas.2012.2279.2282">Fourth order and fourth sum connectivity indices of polyphenylene dendrimers</a>, J. Applied Science, 12 (21), 2012, 2279-2282.
%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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F From _Colin Barker_, May 31 2018: (Start)
%F G.f.: 4*(33 + 104*x) / ((1 - x)*(1 - 2*x)).
%F a(n) = 3*a(n-1) - 2*a(n-2) for n>1.
%F (End)
%p seq(680*2^n-548, n = 0..40);
%o (PARI) Vec(4*(33 + 104*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, May 31 2018
%Y Cf. A305269, A305270, A305272.
%K nonn,easy
%O 0,1
%A _Emeric Deutsch_, May 30 2018
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