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A067771
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Number of vertices in Sierpiński triangle of order n.
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21
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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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listen;
history;
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OFFSET
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0,1
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COMMENTS
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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
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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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FORMULA
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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(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
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EXAMPLE
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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
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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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CROSSREFS
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KEYWORD
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nonn,easy,changed
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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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STATUS
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approved
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