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 A067771 Number of vertices in Sierpiński triangle of order n. 19
 3, 6, 15, 42, 123, 366, 1095, 3282, 9843, 29526, 88575, 265722, 797163, 2391486, 7174455, 21523362, 64570083, 193710246, 581130735, 1743392202, 5230176603, 15690529806, 47071589415, 141214768242, 423644304723, 1270932914166 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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 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 For n>=1, number of edges in a planar Apollonian graph at iteration n. - Andrew D. Walker, Jul 08 2017 REFERENCES Peter Wessendorf and Kristina Downing, personal communication. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..600 Paul Bradley and Peter Rowley, Orbits on k-subsets of 2-transitive Simple Lie-type Groups, 2014. András Kaszanyitzky, Triangular fractal approximating graphs and their covering paths and cycles, arXiv:1710.09475 [math.CO], 2017. See Table 2. C. Lanius, Fractals Eric Weisstein's World of Mathematics, Sierpiński Graph FORMULA a(n) = 3 + 3^1 + 3^2 + 3^3 + 3^4 +...+ 3^n = 3 + Sum_{k=1..n} 3^n. a(0) = 3, a(n) = a(n-1) + 3^n. a(n) = (3/2)*(1+3^n). - Zak Seidov, Mar 19 2007 a(n) = 4*a(n-1) - 3*a(n-2). G.f.: 3*(1-2*x)/((1-x)*(1-3*x)). - Colin Barker, Jan 10 2012 a(n) = A233774(2^n). - Omar E. Pol, Dec 16 2013 a(n) = 3*a(n-1) - 3. - Zak Seidov, Oct 26 2014 MATHEMATICA LinearRecurrence[{4, -3}, {3, 6}, 26] (* or *) CoefficientList[Series[3 (1 - 2 x)/((1 - x) (1 - 3 x)), {x, 0, 25}], x] (* Michael De Vlieger, Feb 02 2017 *) PROG (MAGMA) [(3/2)*(1+3^n): n in [0..30]]; // Vincenzo Librandi, Jun 20 2011 CROSSREFS This is 3*A007051. Cf. A048473. Cf. A003462, A007051, A034472, A024023. - Vladimir Joseph Stephan Orlovsky, Dec 25 2008 Sequence in context: A140824 A001433 A005368 * A289678 A056382 A028401 Adjacent sequences:  A067768 A067769 A067770 * A067772 A067773 A067774 KEYWORD nonn,easy AUTHOR Martin Wessendorf (martinw(AT)mail.ahc.umn.edu), Feb 09 2002 EXTENSIONS More terms from Benoit Cloitre, Feb 22 2002 STATUS approved

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Last modified May 26 19:26 EDT 2019. Contains 323597 sequences. (Running on oeis4.)