A131128 Binomial transform of [1, 1, 5, 1, 5, 1, 5, ...].

1, 2, 8, 20, 44, 92, 188, 380, 764, 1532, 3068, 6140, 12284, 24572, 49148, 98300, 196604, 393212, 786428, 1572860, 3145724, 6291452, 12582908, 25165820, 50331644, 100663292, 201326588, 402653180, 805306364, 1610612732, 3221225468

Offset

0,2

Comments

Row sums of triangle A131129. - Emeric Deutsch, Jun 19 2007

For n >= 4, a(n) is the number of vertices in the dendrimer nanostar NS1[n-3] defined pictorially in the Ashrafi et al. reference (Ns1[3] is shown in Fig. 1) or in the Ahmadi et al. reference (Fig. 1). - Emeric Deutsch, May 17 2018

References

B. Monjardet, Acyclic domains of linear orders: a survey, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 139-160. [From N. J. A. Sloane, Feb 07 2009]

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. B. Ahmadi and M. Sadeghimehr, Atom bond connectivity index of an infinite class NS1[n] of dendrimer nanostars, Optoelectronics and Advanced Materials, 4(7):1040-1042 July 2010.

Ali Reza Ashrafi and Parisa Nikzad, Kekulé index and bounds of energy for nanostar dendrimers, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.

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

Formula

a(n) = 3*2^n - 4 for n >= 1; a(0)=1. Formula follows by replacing [1,1,5,1,5,1,...] with [1,3-2,3+2,3-2,3+2,3-2,...]. - Emeric Deutsch, Jun 19 2007

G.f.: (1 - x + 4x^2)/((1-x)(1-2x)). - Emeric Deutsch, Jul 09 2007

Row sums of triangle A132047. - Gary W. Adamson, Aug 08 2007

a(n) = 2*a(n-1) + 4 for n >= 2, a(0)=1, a(1)=2. - Philippe Deléham, Sep 23 2009

a(n) = 2*A033484(n-1) for n>0. - R. J. Mathar, Feb 27 2019

Example

a(3) = 20 = (1, 3, 3, 1) dot (1, 1, 5, 1) = (1 + 3 + 15 + 1).

Maple

1, seq(3*2^n-4, n = 1 .. 30); # Emeric Deutsch, Jun 19 2007

Mathematica

CoefficientList[Series[(1-x+4x^2)/((1-x)(1-2x)), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 11 2012 *)

Prog

(GAP) Concatenation([1], List([1..30], n->3*2^n-4)); # Muniru A Asiru, May 17 2018

Crossrefs

Cf. A131129, A095121, A132047.

Sequence in context: A009303 A096586 A165751 * A296954 A203604 A240940

Adjacent sequences: A131125 A131126 A131127 * A131129 A131130 A131131

Keyword

nonn,easy

Author

Gary W. Adamson, Jun 16 2007

Status

approved