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a(n) = 68*2^n - 50 (n>=1).
3

%I #26 Jan 23 2025 13:33:35

%S 86,222,494,1038,2126,4302,8654,17358,34766,69582,139214,278478,

%T 557006,1114062,2228174,4456398,8912846,17825742,35651534,71303118,

%U 142606286,285212622,570425294,1140850638,2281701326,4563402702,9126805454,18253610958,36507221966,73014443982,146028888014,292057776078,584115552206,1168231104462

%N a(n) = 68*2^n - 50 (n>=1).

%C a(n) is the first Zagreb index of the nanostar dendrimer NS2[n] from the Madanshekaf et al. reference.

%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 NS2[n] is M(NS2[n]; x,y) = 2*2^n*x*y^2 + (8*2^n - 5)*x^2*y^2 + (6*2^n - 6)*x^2*y^3.

%H Colin Barker, <a href="/A304518/b304518.txt">Table of n, a(n) for n = 1..1000</a>

%H Emeric 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. Madanshekaf and M. Moradi, <a href="http://dx.doi.org/10.22052/ijmc.2014.5541">The first geometric-arithmetic index of some nanostar dendrimers</a>, Iranian J. Math. Chemistry, 5, Supplement 1, 2014, s1-s6.

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

%F From _Colin Barker_, May 15 2018: (Start)

%F G.f.: 2*x*(43 - 18*x) / ((1 - x)*(1 - 2*x)).

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

%p seq(68*2^n-50, n = 1 .. 40);

%t 68*2^Range[50] - 50 (* _Paolo Xausa_, Jul 31 2024 *)

%t LinearRecurrence[{3,-2},{86,222},40] (* _Harvey P. Dale_, Jan 23 2025 *)

%o (GAP) List([1..40],n->68*2^n-50); # _Muniru A Asiru_, May 15 2018

%o (PARI) Vec(2*x*(43 - 18*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, May 15 2018

%Y Cf. A304517, A304519.

%K nonn,easy,changed

%O 1,1

%A _Emeric Deutsch_, May 15 2018