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A223269 T(n,k)=Rolling cube face footprints: number of nXk 0..5 arrays starting with 0 where 0..5 label faces of a cube and every array movement to a horizontal, diagonal or antidiagonal neighbor moves across a corresponding cube edge.

%I A223269 #6 Jul 23 2025 03:56:38
%S A223269 1,4,6,16,48,36,64,576,576,216,256,6144,20992,6912,1296,1024,67584,
%T A223269 622592,765952,82944,7776,4096,737280,19726336,63438848,27951104,
%U A223269 995328,46656,16384,8060928,611319808,5889851392,6467616768,1020002304,11943936
%N A223269 T(n,k)=Rolling cube face footprints: number of nXk 0..5 arrays starting with 0 where 0..5 label faces of a cube and every array movement to a horizontal, diagonal or antidiagonal neighbor moves across a corresponding cube edge.
%C A223269 Table starts
%C A223269 ....1......4.........16...........64.............256...............1024
%C A223269 ....6.....48........576.........6144...........67584.............737280
%C A223269 ...36....576......20992.......622592........19726336..........611319808
%C A223269 ..216...6912.....765952.....63438848......5889851392.......522106961920
%C A223269 .1296..82944...27951104...6467616768...1771674009600....450204914417664
%C A223269 .7776.995328.1020002304.659411697664.534392715870208.389343801904201728
%H A223269 R. H. Hardin, <a href="/A223269/b223269.txt">Table of n, a(n) for n = 1..311</a>
%F A223269 Empirical for column k:
%F A223269 k=1: a(n) = 6*a(n-1)
%F A223269 k=2: a(n) = 12*a(n-1)
%F A223269 k=3: a(n) = 40*a(n-1) -128*a(n-2)
%F A223269 k=4: a(n) = 112*a(n-1) -1024*a(n-2)
%F A223269 k=5: [order 6]
%F A223269 k=6: [order 9]
%F A223269 k=7: [order 19]
%F A223269 Empirical for row n:
%F A223269 n=1: a(n) = 4*a(n-1)
%F A223269 n=2: a(n) = 8*a(n-1) +32*a(n-2)
%F A223269 n=3: a(n) = 24*a(n-1) +256*a(n-2) -1024*a(n-3) for n>4
%F A223269 n=4: [order 6] for n>7
%F A223269 n=5: [order 10] for n>11
%F A223269 n=6: [order 23] for n>24
%e A223269 Some solutions for n=3 k=4
%e A223269 ..0..3..1..2....0..1..0..1....0..4..5..1....0..4..2..4....0..2..1..3
%e A223269 ..0..2..4..3....0..3..5..1....0..4..0..3....0..1..0..4....0..3..4..2
%e A223269 ..4..2..1..2....0..2..0..1....3..1..5..4....3..4..0..1....0..3..4..0
%e A223269 Face neighbors:
%e A223269 0.->.1.2.3.4
%e A223269 1.->.0.2.3.5
%e A223269 2.->.0.1.4.5
%e A223269 3.->.0.1.4.5
%e A223269 4.->.0.3.2.5
%e A223269 5.->.1.3.4.2
%Y A223269 Column 1 is A000400(n-1)
%Y A223269 Column 2 is 4*12^(n-1)
%Y A223269 Column 3 is A223197
%Y A223269 Row 1 is A000302(n-1)
%K A223269 nonn,tabl
%O A223269 1,2
%A A223269 _R. H. Hardin_ Mar 19 2013