OFFSET
0,1
COMMENTS
An order 0 Sierpiński triangle is a triangle. Order n+1 is formed from three copies of order n by identifying pairs of corner vertices of each pair of triangles. - Allan Bickle, Aug 03 2024
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
Vincenzo Librandi, Table of n, a(n) for n = 0..600
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Paul Bradley and Peter Rowley, Orbits on k-subsets of 2-transitive Simple Lie-type Groups, 2014.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
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 Gasket Graph.
Eric Weisstein's World of Mathematics, Total Domination Number.
Index entries for linear recurrences with constant coefficients, signature (4,-3).
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
a(n) = A029858(n+1) + 3. - Allan Bickle, Aug 03 2024
EXAMPLE
Order 0 is a triangle, so a(0) = 3.
Order 1 has three corners (degree 2) and three other vertices, so a(1) = 6.
3 example graphs: o
/ \
o---o
/ \ / \
o o---o---o
/ \ / \ / \
o o---o o---o o---o
/ \ / \ / \ / \ / \ / \ / \
o---o o---o---o o---o---o---o---o
Graph: S_1 S_2 S_3
Vertices: 3 6 15
Edges: 3 9 27
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
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