A305065 a(n) = 48*2^n - 24.

24, 72, 168, 360, 744, 1512, 3048, 6120, 12264, 24552, 49128, 98280, 196584, 393192, 786408, 1572840, 3145704, 6291432, 12582888, 25165800, 50331624, 100663272, 201326568, 402653160, 805306344, 1610612712, 3221225448, 6442450920, 12884901864, 25769803752, 51539607528, 103079215080, 206158430184

Offset

0,1

Comments

a(n) is the number of edges in the dendrimer nanostar G[n], defined pictorially in the Iranmanesh et al. reference (Fig. 1, where G[3] is shown) or in Alikhani et al. reference (Fig. 1, where G[3] is shown).

a(n) is also the number of minimum vertex colorings in the (n+2) X (n+2) king graph. - Eric W. Weisstein, Feb 21 2025

Links

Colin Barker, Table of n, a(n) for n = 0..1000

S. Alikhani, M. A. Iranmanesh, Eccentric connectivity polynomials of an infinite family of dendrimer, Digest J. Nanomaterials and Biostructures, 6 (2011) 253-257.

A. Iranmanesh and N. Dorosti, Computing the Cluj index of a type dendrimer nanostars, MATCH Commun. Math. Comput. Chem. 65, 2011, 209-219.

Eric Weisstein's World of Mathematics, King Graph.

Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.

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

Formula

From Colin Barker, May 25 2018: (Start)

G.f.: 24 / ((1 - x)*(1 - 2*x)).

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

E.g.f.: 24*exp(x)*(2*exp(x) - 1). - Stefano Spezia, Jul 10 2026

Maple

seq(48*2^n - 24, n = 0 .. 40);

Mathematica

48*2^Range[0, 35] - 24 (* Paolo Xausa, Jul 10 2026 *)

Prog

(PARI) Vec(24 / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 25 2018

Crossrefs

Cf. A305064, A305066, A305067.

Sequence in context: A090860 A304374 A064200 * A192833 A292353 A189540

Adjacent sequences: A305062 A305063 A305064 * A305066 A305067 A305068

Keyword

nonn,easy,changed

Author

Emeric Deutsch, May 25 2018

Status

approved