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%I #17 Aug 14 2021 11:49:00
%S 24,147,393,885,1869,3837,7773,15645,31389,62877,125853,251805,503709,
%T 1007517,2015133,4030365,8060829,16121757,32243613,64487325,128974749,
%U 257949597,515899293,1031798685,2063597469,4127195037,8254390173,16508780445,33017560989,66035122077,132070244253,264140488605
%N a(n) = 123*2^n - 99.
%C a(n) is the second Zagreb index of the all-aromatic dendrimer G[n], shown pictorially as DNS1[n] in the Shabani et al. reference (Fig. 1).
%C 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 G[n] is M(G[n]; x, y) = 6*2^n*x^2*y^2 + 12*(2^n - 1)*x^2*y^3 +3* (2^n - 1)*x^3*y^3.
%H Colin Barker, <a href="/A305160/b305160.txt">Table of n, a(n) for n = 0..1000</a>
%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 H. Shabani, A. R. Ashrafi, and I. Gutman, <a href="http://studia.ubbcluj.ro/arhiva/abstract_en.php?editie=CHEMIA&nr=4&an=2010&id_art=8624">Geometric-arithmetic index: an algebraic approach</a>, Studia UBB, Chemia, 55, No. 4, 107-112, 2010.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F From _Colin Barker_, May 30 2018: (Start)
%F G.f.: 3*(8 + 25*x) / ((1 - x)*(1 - 2*x)).
%F a(n) = 3*a(n-1) - 2*a(n-2) for n>1.
%F (End)
%p seq(123*2^n-99, n = 0..40);
%t 123*2^Range[0,40]-99 (* or *) LinearRecurrence[{3,-2},{24,147},40] (* _Harvey P. Dale_, Aug 14 2021 *)
%o (PARI) Vec(3*(8 + 25*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, May 30 2018
%Y Cf. A305158, A305159.
%K nonn,easy
%O 0,1
%A _Emeric Deutsch_, May 29 2018
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