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A067771
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Number of vertices in Sierpiński triangle of order n.
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19
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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Peter Wessendorf and Kristina Downing, personal communication.
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LINKS
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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
Index entries for linear recurrences with constant coefficients, signature (4,-3).
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FORMULA
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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
E.g.f.: 3*(exp(x) + exp(3*x))/2. - Stefano Spezia, Feb 09 2021
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MATHEMATICA
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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 *)
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PROG
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(MAGMA) [(3/2)*(1+3^n): n in [0..30]]; // Vincenzo Librandi, Jun 20 2011
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CROSSREFS
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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 A337326 A056382
Adjacent sequences: A067768 A067769 A067770 * A067772 A067773 A067774
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KEYWORD
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nonn,easy
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AUTHOR
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Martin Wessendorf (martinw(AT)mail.ahc.umn.edu), Feb 09 2002
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EXTENSIONS
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More terms from Benoit Cloitre, Feb 22 2002
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STATUS
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approved
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