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A223197 Rolling cube footprints: number of n X 3 0..5 arrays starting with 0 where 0..5 label faces of a cube and every array movement to a horizontal or vertical neighbor moves across a corresponding cube edge.

%I A223197 #7 Mar 16 2018 06:34:53
%S A223197 16,576,20992,765952,27951104,1020002304,37222350848,1358333739008,
%T A223197 49568888651776,1808888827478016,66010735351693312,
%U A223197 2408891644150546432,87906291641005113344,3207913535188934590464,117064536077508729110528
%N A223197 Rolling cube footprints: number of n X 3 0..5 arrays starting with 0 where 0..5 label faces of a cube and every array movement to a horizontal or vertical neighbor moves across a corresponding cube edge.
%C A223197 Column 3 of A223202.
%H A223197 R. H. Hardin, <a href="/A223197/b223197.txt">Table of n, a(n) for n = 1..210</a>
%F A223197 Empirical: a(n) = 40*a(n-1) - 128*a(n-2).
%F A223197 Conjectures from _Colin Barker_, Mar 16 2018: (Start)
%F A223197 G.f.: 16*x*(1 - 4*x) / (1 - 40*x + 128*x^2).
%F A223197 a(n) = ((20-4*sqrt(17))^n*(-3+sqrt(17)) + (3+sqrt(17))*(4*(5+sqrt(17)))^n) / (4*sqrt(17)).
%F A223197 (End)
%e A223197 Some solutions for n=3:
%e A223197 ..0..1..0....0..1..5....0..3..4....0..4..5....0..3..4....0..1..5....0..3..1
%e A223197 ..2..0..3....2..5..2....1..0..2....1..5..1....3..0..2....1..2..1....3..4..3
%e A223197 ..0..2..0....5..1..5....5..2..1....5..1..2....4..2..4....0..4..0....4..3..1
%e A223197 Face neighbors:
%e A223197 0,5 -> 1 2 3 4
%e A223197 1,4 -> 0 2 3 5
%e A223197 2,3 -> 0 1 4 5
%Y A223197 Cf. A223202.
%K A223197 nonn
%O A223197 1,1
%A A223197 _R. H. Hardin_, Mar 18 2013