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A304167 a(n) = 3^n - 2^(n-1) + 2 (n>=1).

%I A304167 #24 Jul 24 2022 12:21:53
%S A304167 4,9,25,75,229,699,2125,6435,19429,58539,176125,529395,1590229,
%T A304167 4774779,14332525,43013955,129074629,387289419,1161999325,3486260115,
%U A304167 10459304629,31378962459,94138984525,282421147875,847271832229,2541832273899,7625530376125,22876658237235,68630108929429,205890595223739
%N A304167 a(n) = 3^n - 2^(n-1) + 2 (n>=1).
%C A304167 For n>=2, a(n) is the number of vertices of the Sierpinski Gasket Rhombus graph SR(n) (see Theorem 2.1 in the D. Antony Xavier et al. reference).
%H A304167 Colin Barker, <a href="/A304167/b304167.txt">Table of n, a(n) for n = 1..1000</a>
%H A304167 D. Antony Xavier, M. Rosary, and Andrew Arokiaraj, <a href="https://www.ijmsc.com/index.php/ijmsc/article/view/261">Topological properties of Sierpinski Gasket Rhombus graphs</a>, International J. of Mathematics and Soft Computing, 4, No. 2, 2014, 95-104.
%H A304167 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F A304167 From _Colin Barker_, May 10 2018: (Start)
%F A304167 G.f.: x*(4 - 15*x + 15*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
%F A304167 a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>3.
%F A304167 (End)
%F A304167 a(n) = A083313(n)+2. - _R. J. Mathar_, Jul 24 2022
%p A304167 seq(3^n-2^(n-1)+2, n = 1 .. 40);
%o A304167 (PARI) Vec(x*(4 - 15*x + 15*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ _Colin Barker_, May 10 2018
%o A304167 (GAP) List([1..40],n->3^n-2^(n-1)+2); # _Muniru A Asiru_, May 10 2018
%Y A304167 Cf. A304168, A304169, A304170.
%K A304167 nonn,easy
%O A304167 1,1
%A A304167 _Emeric Deutsch_, May 10 2018