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

%I A298202 #26 Jun 15 2025 20:56:12
%S A298202 1,16,102400,40823664148480000,
%T A298202 4024143600922674552523331296813921054228480000000000
%N A298202 Number of Eulerian cycles in the n-Sierpinski gasket graph.
%C A298202 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.
%C A298202 Different starting points and directions do not make two circuits distinct. - _Allan Bickle_, Aug 06 2024
%C A298202 a(6) has 157 decimal digits. - _Andrew Howroyd_, Sep 10 2019
%H A298202 Andrew Howroyd, <a href="/A298202/b298202.txt">Table of n, a(n) for n = 1..7</a>
%H A298202 A. Hinz, S. Klavzar, and S. Zemljic, <a href="https://doi.org/10.1016/j.dam.2016.09.024">A survey and classification of Sierpinski-type graphs</a>, Discrete Applied Mathematics 217 3 (2017), 565-600.
%H A298202 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerianCycle.html">Eulerian Cycle</a>.
%H A298202 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiGasketGraph.html">Sierpinski Gasket Graph</a>.
%e A298202 3 example graphs:                        o
%e A298202                                         / \
%e A298202                                        o---o
%e A298202                                       / \ / \
%e A298202                         o            o---o---o
%e A298202                        / \          / \     / \
%e A298202              o        o---o        o---o   o---o
%e A298202             / \      / \ / \      / \ / \ / \ / \
%e A298202            o---o    o---o---o    o---o---o---o---o
%e A298202 Graph:      S_1        S_2              S_3
%e A298202 A triangle has a single Eulerian circuit, so a(1) = 1.
%e A298202 The level 2 graph has 16 distinct circuits, 12 that reverse at a middle vertex and 4 that don't, so a(2) = 16.
%t A298202 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 *)
%o A298202 (PARI)
%o A298202 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]}
%o A298202 a(n)={my(u=[1,0,0]); for(n=2, n, u=P(u)); u[1]} \\ _Andrew Howroyd_, Sep 12 2019
%Y A298202 Cf. A246959.
%K A298202 nonn
%O A298202 1,2
%A A298202 _Eric W. Weisstein_, Jan 14 2018
%E A298202 a(4)-a(5) from _Andrew Howroyd_, Sep 10 2019