A304515 - OEIS

%I #18 May 16 2018 11:32:26

%S 96,414,1050,2322,4866,9954,20130,40482,81186,162594,325410,651042,

%T 1302306,2604834,5209890,10420002,20840226,41680674,83361570,

%U 166723362,333446946,666894114,1333788450,2667577122,5335154466,10670309154,21340618530,42681237282,85362474786,170724949794,341449899810,682899799842,1365799599906,2731599200034,5463198400290,10926396800802,21852793601826,43705587203874

%N a(n) = 159*2^n - 222 (n>=1).

%C a(n) is the first Zagreb index of the nanostar dendrimer D[n] from the Ghorbani 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 D[n] is M(D[n]; x,y) = 12*(2^n - 1)*x^2*y^2 + 3*(5*2^n - 8)*x^2*y^3 + 3*(2*2^n - 3)*x^3*y^3.

%H Colin Barker, <a href="/A304515/b304515.txt">Table of n, a(n) for n = 1..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. Ghorbani and M. Songhori, <a href="http://dx.doi.org/10.22052/ijmc.2010.5155">Some topological indices of nanostar dendrimers</a>, Iranian J. Math. Chemistry, 1, No. 2, 2010, 57-65.

%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.: 6*x*(16 + 21*x) / ((1 - x)*(1 - 2*x)).

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

%F (End)

%p seq(159*2^n-222, n = 1 .. 40);

%t Rest@ CoefficientList[Series[6 x (16 + 21 x)/((1 - x) (1 - 2 x)), {x, 0, 38}], x] (* or *)

%t LinearRecurrence[{3, -2}, {96, 414}, 38] (* or *)

%t Array[159*2^# - 222 &, 38] (* _Michael De Vlieger_, May 15 2018 *)

%o (GAP) List([1..40],n->159*2^n-222); # _Muniru A Asiru_, May 15 2018

%o (PARI) Vec(6*x*(16 + 21*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, May 15 2018

%Y Cf. A304513, A304514, A304516.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, May 15 2018