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%I #39 Feb 27 2019 04:53:27
%S 1,2,8,20,44,92,188,380,764,1532,3068,6140,12284,24572,49148,98300,
%T 196604,393212,786428,1572860,3145724,6291452,12582908,25165820,
%U 50331644,100663292,201326588,402653180,805306364,1610612732,3221225468
%N Binomial transform of [1, 1, 5, 1, 5, 1, 5, ...].
%C Row sums of triangle A131129. - _Emeric Deutsch_, Jun 19 2007
%C 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
%D 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]
%H Vincenzo Librandi, <a href="/A131128/b131128.txt">Table of n, a(n) for n = 0..1000</a>
%H M. B. Ahmadi and M. Sadeghimehr, <a href="https://oam-rc.inoe.ro/download.php?idu=1158=52">Atom bond connectivity index of an infinite class NS1[n] of dendrimer nanostars</a>, Optoelectronics and Advanced Materials, 4(7):1040-1042 July 2010.
%H Ali Reza Ashrafi and Parisa Nikzad, <a href="http://www.chalcogen.ro/383_Ashrafi.pdf">Kekulé index and bounds of energy for nanostar dendrimers</a>, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F 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
%F G.f.: (1 - x + 4x^2)/((1-x)(1-2x)). - _Emeric Deutsch_, Jul 09 2007
%F Row sums of triangle A132047. - _Gary W. Adamson_, Aug 08 2007
%F a(n) = 2*a(n-1) + 4 for n >= 2, a(0)=1, a(1)=2. - _Philippe Deléham_, Sep 23 2009
%F a(n) = 2*A033484(n-1) for n>0. - _R. J. Mathar_, Feb 27 2019
%e a(3) = 20 = (1, 3, 3, 1) dot (1, 1, 5, 1) = (1 + 3 + 15 + 1).
%p 1, seq(3*2^n-4, n = 1 .. 30); # _Emeric Deutsch_, Jun 19 2007
%t CoefficientList[Series[(1-x+4x^2)/((1-x)(1-2x)),{x,0,40}],x] (* _Vincenzo Librandi_, Apr 11 2012 *)
%o (GAP) Concatenation([1],List([1..30], n->3*2^n-4)); # _Muniru A Asiru_, May 17 2018
%Y Cf. A131129, A095121, A132047.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Jun 16 2007
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