

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
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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 ksubsets of 2transitive Simple Lietype 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(n1) + 3^n. a(n) = (3/2)*(1+3^n).  Zak Seidov, Mar 19 2007
a(n) = 4*a(n1)  3*a(n2).
G.f.: 3*(12*x)/((1x)*(13*x)).  Colin Barker, Jan 10 2012
a(n) = A233774(2^n).  Omar E. Pol, Dec 16 2013
a(n) = 3*a(n1)  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



