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Rolling cube footprints: number of n X 5 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.
1

%I #8 Aug 19 2018 09:36:49

%S 81,9261,1059723,121264857,13876429707,1587890407761,181703507374179,

%T 20792470582897209,2379298227030964827,272264906211251105313,

%U 31155480347969275662483,3565145318286297427548489

%N Rolling cube footprints: number of n X 5 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 Column 5 of A223331.

%H R. H. Hardin, <a href="/A223328/b223328.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 117*a(n-1) - 294*a(n-2).

%F Empirical g.f.: 27*x*(3 - 8*x) / (1 - 117*x + 294*x^2). - _Colin Barker_, Aug 19 2018

%e Some solutions for n=3:

%e ..0..2..0..2..6....0..4..6..4..0....0..4..5..7..6....0..4..0..2..3

%e ..0..2..0..2..6....6..2..0..4..0....0..4..6..4..6....6..2..6..2..0

%e ..0..2..6..4..0....0..2..0..1..3....0..2..0..2..0....0..2..3..2..0

%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