A304607 - OEIS

%I #30 May 30 2026 16:40:47

%S 644,1148,2156,4172,8204,16268,32396,64652,129164,258188,516236,

%T 1032332,2064524,4128908,8257676,16515212,33030284,66060428,132120716,

%U 264241292,528482444,1056964748,2113929356,4227858572,8455717004,16911433868,33822867596,67645735052,135291469964,270582939788,541165879436,1082331758732

%N a(n) = 252*2^n + 140 (n>=1).

%C a(n) is the first Zagreb index of the nanostar dendrimer G[n] from the Ashrafi 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 G[n] is M(G[n]; x,y) = 4*x*y^4 + (18*2^n + 21)*x^2*y^2 + (36*2^n - 9)*x^2*y^3 + 3*x^2*y^4 + 9*x^3*y^4.

%H Colin Barker, <a href="/A304607/b304607.txt">Table of n, a(n) for n = 1..1000</a>

%H A. R. Ashrafi, A. Karbasioun, and M. V. Diudea, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match65/n1/match65n1_193-200.pdf">Computing Wiener and detour indices of a new type of nanostar dendrimers</a>, MATCH Commun. Math. Comput. Chem. 65, 2011, 193-200.

%H E. Deutsch and Sandi Klavzar, <a href="https://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 _Michael De Vlieger_, May 15 2018: (Start)

%F G.f.: 28*x*(23 - 28*x)/(1 - 3*x + 2*x^2).

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

%p seq(252*2^n+140, n = 1 .. 40);

%t CoefficientList[Series[28*(23 - 28*x)/(1 - 3*x + 2*x^2), {x, 0, 31}], x] (* or *)

%t LinearRecurrence[{3, -2}, {644, 1148}, 32] (* or *)

%t Array[252*2^# + 140 &, 32] (* _Michael De Vlieger_, May 15 2018 *)

%o (PARI) a(n) = 252*2^n + 140; \\ _Altug Alkan_, May 15 2018

%o (PARI) Vec(28*x*(23 - 28*x)/(1 - 3*x + 2*x^2) + O(x^40)) \\ _Colin Barker_, May 23 2018

%o (GAP) List([1..40],n->252*2^n+140); # _Muniru A Asiru_, May 16 2018

%Y Cf. A304605, A304606, A304608.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, May 15 2018