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a(n) = 126*2^n - 22 (n>=1).
2

%I #23 May 22 2018 09:48:19

%S 230,482,986,1994,4010,8042,16106,32234,64490,129002,258026,516074,

%T 1032170,2064362,4128746,8257514,16515050,33030122,66060266,132120554,

%U 264241130,528482282,1056964586,2113929194,4227858410,8455716842,16911433706,33822867434,67645734890,135291469802,270582939626,541165879274

%N a(n) = 126*2^n - 22 (n>=1).

%C a(n) is the first Zagreb index of the dendrimer nanostar NS1[n], defined pictorially in the Ashrafi et al. reference (Ns1[3] is shown in Fig. 1) or in the Ahmadi et al. reference (Fig. 1).

%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 NS1[n] is M(NS1[n]; x,y) = xy^4 + (9*2^n + 3)x^2*y^2 + (18*2^n - 12)x^2*y^3 + 3x^3*y^4 .

%H Colin Barker, <a href="/A304389/b304389.txt">Table of n, a(n) for n = 1..1000</a>

%H M. B. Ahmadi and M. Sadeghimehr, <a href="https://oam-rc.inoe.ro/download.php?idu=1158=52">Atom bond connectivity index of an infinite class NS1[n] of dendrimer nanostars</a>, Optoelectronics and Advanced Materials, 4(7):1040-1042 July 2010.

%H Ali Reza Ashrafi and Parisa Nikzad, <a href="http://www.chalcogen.ro/383_Ashrafi.pdf">Kekulé index and bounds of energy for nanostar dendrimers</a>, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.

%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 18 2018: (Start)

%F G.f.: 2*x*(115 - 104*x) / ((1 - x)*(1 - 2*x)).

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

%F (End)

%p seq(126*2^n-22, n = 1 .. 40);

%o (PARI) a(n) = 126*2^n - 22; \\ _Altug Alkan_, May 13 2018

%o (PARI) Vec(2*x*(115 - 104*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, May 18 2018

%o (GAP) List([1..40],n->126*2^n-22); # _Muniru A Asiru_, May 13 2018

%Y Cf. A304386, A304387, A304388.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, May 13 2018