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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
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
REFERENCES
Peter Wessendorf and Kristina Downing, personal communication.
LINKS
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, Dorogovtsev-Goltsev-Mendes Graph.
Eric Weisstein's World of Mathematics, Sierpiński Graph.
Eric Weisstein's World of Mathematics, Total Domination Number.
FORMULA
a(n) = (3/2)*(3^n + 1).
a(n) = 3 + 3^1 + 3^2 + 3^3 + 3^4 + ... + 3^n = 3 + Sum_{k=1..n} 3^n.
a(n) = 3*A007051(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
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 *)
Table[3/2 (3^n + 1), {n, 0, 20}] (* Eric W. Weisstein, Jan 14 2024 *)
3/2 (3^Range[0, 20] + 1) (* Eric W. Weisstein, Jan 14 2024 *)
PROG
(Magma) [(3/2)*(1+3^n): n in [0..30]]; // Vincenzo Librandi, Jun 20 2011
CROSSREFS
Cf. A048473.
Sequence in context: A140824 A001433 A005368 * A289678 A337326 A056382
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 March 28 11:46 EDT 2024. Contains 371241 sequences. (Running on oeis4.)