A305263 - OEIS

%I #9 Sep 03 2022 23:52:14

%S 58,738,2098,4818,10258,21138,42898,86418,173458,347538,695698,

%T 1392018,2784658,5569938,11140498,22281618,44563858,89128338,

%U 178257298,356515218,713031058,1426062738,2852126098,5704252818,11408506258,22817013138,45634026898,91268054418,182536109458,365072219538,730144439698

%N a(n) = 680*2^n - 622.

%C a(n) is the first Zagreb index of the nanostar dendrimer G[n], shown pictorially as NSD[n] in the Rostami et al. reference (Fig. 2).

%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 nanostar dendrimer G[n] is M(G[n];x,y) = (56*2^n - 48)*x^2*y^2 + (48*2^n - 44)*x^2*y^3 +(36* 2^n - 35)*x^3*y^3.

%H Colin Barker, <a href="/A305263/b305263.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 M. Rostami, M. Shabanian, and H. Moghanian, <a href="http://www.chalcogen.ro/247_Rostami.pdf">Some topological indices for theoretical study of two types of nanostar dendrimers</a>, Digest J. of Nanomaterials and Biostructures, 7, No. 1, 2012, 247-252.

%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.: 2*(29 + 282*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-622, n = 0..40);

%t CoefficientList[Series[2*(29 + 282*x)/((1 - x)*(1 - 2*x)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Sep 03 2022 *)

%o (PARI) Vec(2*(29 + 282*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, May 31 2018

%Y Cf. A305261, A305262, A305264.

%K nonn,easy

%O 0,1

%A _Emeric Deutsch_, May 29 2018