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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 February 18 23:26 EST 2018. Contains 299330 sequences. (Running on oeis4.)