OFFSET
2,1
COMMENTS
A level 1 Sierpiński triangle graph is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.
Antipodal vertices are a pair of vertices at maximum distance in a graph. The diameter of the level n Sierpiński triangle graph is 2^(n-1).
LINKS
Paolo Xausa, Table of n, a(n) for n = 2..1000
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021), pp. 295-303.
Andreas M. Hinz, Sandi Klavžar, and Sara Sabrina Zemljič, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics, Volume 217, Part 3 (2017), pages 565-600.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Index entries for linear recurrences with constant coefficients, signature (6,-8).
FORMULA
a(n) = 3*2^(n-3)*(2^(n-2)+3).
a(n) = 3*A257273(n-2).
a(n) = A375256(n-1) + 3.
From Elmo R. Oliveira, Mar 15 2026: (Start)
G.f.: 3*x^2*(-7*x + 2)/((2*x - 1)*(4*x - 1)).
E.g.f.: (1/32)*(3*exp(4*x) + 36*exp(2*x) - 84*x - 39).
a(n) = 6*a(n-1) - 8*a(n-2) for n >= 4. (End)
EXAMPLE
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
For S_2, there are 3 pairs of corners and 3 pairs of a corner and a middle vertex, so a(2) = 6.
MATHEMATICA
A370933[n_] := 3*2^(n - 3)*(2^(n - 2) + 3);
Array[A370933, 30, 2] (* or *)
LinearRecurrence[{6, -8}, {6, 15}, 30] (* Paolo Xausa, Sep 23 2024 *)
PROG
(PARI) a(n) = 3*2^(n-3)*(2^(n-2)+3); \\ Michel Marcus, Aug 08 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Allan Bickle, Aug 07 2024
EXTENSIONS
More terms from Michel Marcus, Aug 08 2024
STATUS
approved