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%I #82 Jan 14 2024 09:00:12
%S 3,6,15,42,123,366,1095,3282,9843,29526,88575,265722,797163,2391486,
%T 7174455,21523362,64570083,193710246,581130735,1743392202,5230176603,
%U 15690529806,47071589415,141214768242,423644304723,1270932914166
%N Number of vertices in Sierpiński triangle of order n.
%C This sequence represents another link from the product factor space Q X Q / {(1,1), (-1, -1)} to Sierpiński's triangle. The first "link" found was to sequence A048473. - _Creighton Dement_, Aug 05 2004
%C a(n) equals the number of orbits of the finite group PSU(3,3^n) on subsets of size 3 of the 3^(3n)+1 isotropic points of a unitary 3 space. - _Paul M. Bradley_, Jan 31 2017
%C For n >= 1, number of edges in a planar Apollonian graph at iteration n. - _Andrew D. Walker_, Jul 08 2017
%C Also the total domination number of the (n+3)-Dorogovtsev-Goltsev-Mendes graph, using the convention DGM(0) = P_2. - _Eric W. Weisstein_, Jan 14 2024
%D Peter Wessendorf and Kristina Downing, personal communication.
%H Vincenzo Librandi, <a href="/A067771/b067771.txt">Table of n, a(n) for n = 0..600</a>
%H Paul Bradley and Peter Rowley, <a href="http://eprints.ma.man.ac.uk/2167/01/covered/MIMS_ep2014_42.pdf">Orbits on k-subsets of 2-transitive Simple Lie-type Groups</a>, 2014.
%H András Kaszanyitzky, <a href="https://arxiv.org/abs/1710.09475">Triangular fractal approximating graphs and their covering paths and cycles</a>, arXiv:1710.09475 [math.CO], 2017. See Table 2.
%H C. Lanius, <a href="http://math.rice.edu/~lanius/fractals/">Fractals</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Dorogovtsev-Goltsev-MendesGraph.html">Dorogovtsev-Goltsev-Mendes Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiGraph.html">Sierpiński Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominationNumber.html">Total Domination Number</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F a(n) = (3/2)*(3^n + 1).
%F a(n) = 3 + 3^1 + 3^2 + 3^3 + 3^4 + ... + 3^n = 3 + Sum_{k=1..n} 3^n.
%F a(n) = 3*A007051(n).
%F a(0) = 3, a(n) = a(n-1) + 3^n. a(n) = (3/2)*(1+3^n). - _Zak Seidov_, Mar 19 2007
%F a(n) = 4*a(n-1) - 3*a(n-2).
%F G.f.: 3*(1-2*x)/((1-x)*(1-3*x)). - _Colin Barker_, Jan 10 2012
%F a(n) = A233774(2^n). - _Omar E. Pol_, Dec 16 2013
%F a(n) = 3*a(n-1) - 3. - _Zak Seidov_, Oct 26 2014
%F E.g.f.: 3*(exp(x) + exp(3*x))/2. - _Stefano Spezia_, Feb 09 2021
%t LinearRecurrence[{4, -3}, {3, 6}, 26] (* or *)
%t CoefficientList[Series[3 (1 - 2 x)/((1 - x) (1 - 3 x)), {x, 0, 25}], x] (* _Michael De Vlieger_, Feb 02 2017 *)
%t Table[3/2 (3^n + 1), {n, 0, 20}] (* _Eric W. Weisstein_, Jan 14 2024 *)
%t 3/2 (3^Range[0, 20] + 1) (* _Eric W. Weisstein_, Jan 14 2024 *)
%o (Magma) [(3/2)*(1+3^n): n in [0..30]]; // _Vincenzo Librandi_, Jun 20 2011
%Y Cf. A048473.
%Y Cf. A003462, A007051, A034472, A024023. - _Vladimir Joseph Stephan Orlovsky_, Dec 25 2008
%K nonn,easy
%O 0,1
%A Martin Wessendorf (martinw(AT)mail.ahc.umn.edu), Feb 09 2002
%E More terms from _Benoit Cloitre_, Feb 22 2002
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