OFFSET
1,2
COMMENTS
A level 1 Sierpiński triangle 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.
Different starting points and directions do not make two circuits distinct. - Allan Bickle, Aug 06 2024
a(6) has 157 decimal digits. - Andrew Howroyd, Sep 10 2019
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..7
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Eric Weisstein's World of Mathematics, Eulerian Cycle
Eric Weisstein's World of Mathematics, Sierpinski Sieve Graph
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
A triangle has a single Eulerian circuit, so a(1) = 1.
The level 2 graph has 16 distinct circuits, 12 that reverse at a middle vertex and 4 that don't, so a(2) = 16.
MATHEMATICA
NestList[Function[{e, f, g}, {16 e^3 + 48 f e^2, 3 e^3 + (32 f + 8 g) e^2 + 56 f^2 e, e^3 + (30 f + 12 g) e^2 + (156 f^2 + 96 g f) e + 112 f^3}] @@ # &, {1, 0, 0}, 5][[All, 1]] (* Eric W. Weisstein, Feb 02 2024 based on code from Andrew Howroyd *)
PROG
(PARI)
P(u)={my([e, f, g]=u); [16*e^3 + 48*f*e^2, 3*e^3 + (32*f + 8*g)*e^2 + 56*f^2*e, e^3 + (30*f + 12*g)*e^2 + (156*f^2 + 96*g*f)*e + 112*f^3]}
a(n)={my(u=[1, 0, 0]); for(n=2, n, u=P(u)); u[1]} \\ Andrew Howroyd, Sep 12 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jan 14 2018
EXTENSIONS
a(4)-a(5) from Andrew Howroyd, Sep 10 2019
STATUS
approved