%I #10 Aug 19 2018 10:42:02
%S 64,243,3969,64827,1059723,17324685,283231809,4630406067,75700050363,
%T 1237579942725,20232537609249,330771018512907,5407599817782603,
%U 88405979220238365,1445302430884160289,23628482316968449347
%N Rolling cube footprints: number of 3 X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.
%C Row 3 of A223331.
%H R. H. Hardin, <a href="/A223332/b223332.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 18*a(n-1) - 27*a(n-2) for n>4.
%F Conjectures from _Colin Barker_, Aug 19 2018: (Start)
%F G.f.: x*(64 - 909*x + 1323*x^2 - 54*x^3) / (1 - 18*x + 27*x^2).
%F a(n) = (49/54)*((9-3*sqrt(6))^n + (3*(3+sqrt(6)))^n) for n>2.
%F (End)
%e Some solutions for n=3:
%e 0 2 6 0 4 6 0 2 6 0 1 5 0 2 3 0 1 5 0 2 6
%e 6 7 6 6 2 3 3 7 5 5 4 6 3 7 6 5 7 3 0 4 0
%e 3 7 6 6 2 0 6 4 6 6 2 0 6 7 3 6 7 6 6 4 6
%e Vertex neighbors:
%e 0 -> 1 2 4
%e 1 -> 0 3 5
%e 2 -> 0 3 6
%e 3 -> 1 2 7
%e 4 -> 0 5 6
%e 5 -> 1 4 7
%e 6 -> 2 4 7
%e 7 -> 3 5 6
%Y Cf. A223331.
%K nonn
%O 1,1
%A _R. H. Hardin_, Mar 19 2013
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