A298202 Number of Eulerian cycles in the n-Sierpinski gasket graph.

1, 16, 102400, 40823664148480000, 4024143600922674552523331296813921054228480000000000

Offset: 1

Author

Eric W. Weisstein, Jan 14 2018

Keywords

• nonn

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

Examples

			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.
		

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 Gasket Graph.

Crossrefs

Cf. A246959.

Programs

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 *)

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

Extensions

a(4)-a(5) from Andrew Howroyd, Sep 10 2019