A227490 The Wiener index of the nanostar dendrimer D_n, defined pictorially in the Ghorbani et al. references and recursively in the Deutsch et al. reference.

666, 35325, 412443, 3099663, 19013175, 104225031, 533102247, 2604327399, 12319287399, 56913753447, 258258898791, 1155566158695, 5112617020263, 22412337970023, 97497752214375, 421386451835751, 1811131262622567, 7746874238425959

Offset

1,1

Comments

a(1) has been checked by the direct computation of the Wiener index (using Maple).

In the Ghorbani & Songhori reference the formula for the Wiener index (p. 62) contains some errors.

The Deutsch et al. reference contains also the Hosoya polynomial of D_n.

Links

Muniru A Asiru, Table of n, a(n) for n = 1..500

E. Deutsch, S. Klavzar, Computing the Hosoya polynomial of graphs from primary subgraphs, MATCH Commun. Math. Comput. Chem., 70, 2013, 627-644.

M. Ghorbani, A. Mohammadi, F. Madadi, Some topological indices of nanostar dendrimers, Optoelectronics and Adv. Materials - Rapid Comm., 4, 2010, 1871-1873.

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 (11,-42,64,-32).

Formula

a(n) = - 9369 + 56205*2^(n-1) - 75411*2^(2n-2) + 29241*2^(2n-2)*n.

G.f.: 9*x*(74 + 3111*x + 5760*x^2 + 424*x^3)/((1-x)*(1-2*x)*(1-4*x)^2).

a(n) = 11*a(n-1) - 42*a(n-2) + 64*a(n-3) - 32*a(n-4) for n>4. - Colin Barker, Jun 15 2018

Maple

a := proc (n) options operator, arrow: -9369+56205*2^(n-1)-75411*2^(2*n-2)+29241*2^(2*n-2)*n end proc: seq(a(n), n = 1 .. 22);

Mathematica

Table[-9369+56205*2^(n-1)-75411*2^(2n-2)+29241*2^(2n-2) n, {n, 20}] (* or *) LinearRecurrence[{11, -42, 64, -32}, {666, 35325, 412443, 3099663}, 20] (* Harvey P. Dale, Dec 24 2017 *)

Prog

(GAP) List([1..20], n-> 29241*2^(2*n-2)*n-75411*2^(2*n-2)+56205*2^(n-1)-9369); # Muniru A Asiru, Jun 15 2018
(PARI) Vec(9*x*(74 + 3111*x + 5760*x^2 + 424*x^3)/((1-x)*(1-2*x)*(1-4*x)^2) + O(x^40)) \\ Colin Barker, Jun 15 2018

Crossrefs

Cf. A227491.

Sequence in context: A104181 A221014 A057564 * A171114 A328551 A172922

Adjacent sequences: A227487 A227488 A227489 * A227491 A227492 A227493

Keyword

nonn,easy

Author

Emeric Deutsch, Jul 16 2013

Status

approved