A304515 a(n) = 159*2^n - 222 (n>=1).

96, 414, 1050, 2322, 4866, 9954, 20130, 40482, 81186, 162594, 325410, 651042, 1302306, 2604834, 5209890, 10420002, 20840226, 41680674, 83361570, 166723362, 333446946, 666894114, 1333788450, 2667577122, 5335154466, 10670309154, 21340618530, 42681237282, 85362474786, 170724949794, 341449899810, 682899799842, 1365799599906, 2731599200034, 5463198400290, 10926396800802, 21852793601826, 43705587203874

Offset

1,1

Comments

a(n) is the first Zagreb index of the nanostar dendrimer D[n] from the Ghorbani et al. reference.

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.

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.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.

M. Ghorbani and M. Songhori, Some topological indices of nanostar dendrimers, Iranian J. Math. Chemistry, 1, No. 2, 2010, 57-65.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Formula

From Colin Barker, May 15 2018: (Start)

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

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

(End)

Maple

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

Mathematica

Rest@ CoefficientList[Series[6 x (16 + 21 x)/((1 - x) (1 - 2 x)), {x, 0, 38}], x] (* or *)
LinearRecurrence[{3, -2}, {96, 414}, 38] (* or *)
Array[159*2^# - 222 &, 38] (* Michael De Vlieger, May 15 2018 *)

Prog

(GAP) List([1..40], n->159*2^n-222); # Muniru A Asiru, May 15 2018
(PARI) Vec(6*x*(16 + 21*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 15 2018

Crossrefs

Cf. A304513, A304514, A304516.

Sequence in context: A292543 A051465 A179825 * A233818 A233811 A233717

Adjacent sequences: A304512 A304513 A304514 * A304516 A304517 A304518

Keyword

nonn,easy

Author

Emeric Deutsch, May 15 2018

Status

approved