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A278130 a(n) = 492*2^n - 222. 1
270, 762, 1746, 3714, 7650, 15522, 31266, 62754, 125730, 251682, 503586, 1007394, 2015010, 4030242, 8060706, 16121634, 32243490, 64487202, 128974626, 257949474, 515899170, 1031798562, 2063597346, 4127194914, 8254390050, 16508780322, 33017560866, 66035121954, 132070244130 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) is the first Zagreb index of the phenylacetylene dendrimer NSB[n] defined pictorially in the Yarahmadi references. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph.

The M-polynomial of the dendrimer NSB[n] is M(NSB[n],x,y) = 9*2^n*x*y^4 + (24*2^n - 12)*x^2*y^2 + (48*2^n - 24)*x^2*y^3 + (15*2^n - 9)*x^3*y^3 + 3*2^n*x^3*y^4.

From Bruno Berselli, Nov 16 2016: (Start)

In general, this type of formula b(n) = k*2^n - k (where n>=0 and h, k are given constants) has:

O.g.f.: (h - k - (h - 2*k)*x)/((1 - x)*(1 - 2*x));

E.g.f.: (-k + h*exp(x))*exp(x);

Linear recurrence: b(n) = 3*b(n-1) - 2*b(n-2);

Signature of the recurrence: (3,-2). (End)

LINKS

Table of n, a(n) for n=0..28.

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

I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.

Z. Yarahmadi, Eccentric connectivity and augmented eccentric connectivity indices of N-branches phenylacetylenes nanostar dendrimers, Iranian J. Math. Chem., 1, No. 2, 2010, 105-110.

Z. Yarahmadi and G. H. Fath-Tabar, The Wiener, Szeged, PI, Vertex PI, the first and second Zagreb indices of N-branched phenylacetylenes dendrimers,  MATCH: Commun. Math. Comput. Chem, 65 (2011)  201-208.

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

FORMULA

O.g.f.: 6*(45 - 8*x)/((1 - x)*(1 - 2*x)).

E.g.f.: 6*(-37 + 82*exp(x))*exp(x).

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

MAPLE

seq(492*2^n-222, n=0..35)

MATHEMATICA

Table[492 2^n - 222, {n, 0, 35}] (* Vincenzo Librandi, Nov 16 2016 *)

PROG

(MAGMA) [492*2^n-222: n in [0..35]]; // Vincenzo Librandi, Nov 16 2016

CROSSREFS

Cf. A278131.

Sequence in context: A291789 A292766 A180151 * A206088 A029770 A028529

Adjacent sequences:  A278127 A278128 A278129 * A278131 A278132 A278133

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Nov 15 2016

EXTENSIONS

Edited by Bruno Berselli, Nov 16 2016

STATUS

approved

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Last modified October 23 16:46 EDT 2019. Contains 328373 sequences. (Running on oeis4.)