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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 or vertical neighbor moves across a corresponding cube edge
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%I #4 Mar 18 2013 06:16:50

%S 1,4,4,16,48,16,64,576,576,64,256,6912,20992,6912,256,1024,82944,

%T 765952,765952,82944,1024,4096,995328,27951104,85327872,27951104,

%U 995328,4096,16384,11943936,1020002304,9515827200,9515827200,1020002304,11943936

%N 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 or vertical neighbor moves across a corresponding cube edge

%C Table starts

%C ....1......4.........16............64..............256................1024

%C ....4.....48........576..........6912............82944..............995328

%C ...16....576......20992........765952.........27951104..........1020002304

%C ...64...6912.....765952......85327872.......9515827200.......1061444124672

%C ..256..82944...27951104....9515827200....3249109204992....1110327429169152

%C .1024.995328.1020002304.1061444124672.1110327429169152.1163614255186968576

%H R. H. Hardin, <a href="/A223202/b223202.txt">Table of n, a(n) for n = 1..364</a>

%F Empirical for column k:

%F k=1: a(n) = 4*a(n-1)

%F k=2: a(n) = 12*a(n-1)

%F k=3: a(n) = 40*a(n-1) -128*a(n-2)

%F k=4: a(n) = 144*a(n-1) -3840*a(n-2) +24576*a(n-3)

%F k=5: a(n) = 512*a(n-1) -66560*a(n-2) +3014656*a(n-3) -50331648*a(n-4) +268435456*a(n-5)

%F k=6: [order 7]

%F k=7: [order 13]

%e Some solutions for n=3 k=4

%e ..0..2..4..2....0..1..5..4....0..3..4..2....0..2..0..4....0..2..0..3

%e ..3..1..2..5....3..5..3..0....3..1..3..5....3..5..2..0....3..5..2..5

%e ..0..3..1..2....0..2..0..2....0..3..5..4....1..3..5..2....1..2..1..2

%e Face neighbors:

%e 0,5 -> 1 2 3 4

%e 1,4 -> 0 2 3 5

%e 2,3 -> 0 1 4 5

%Y Column 1 is A000302(n-1)

%Y Column 2 is 4*12^(k-1)

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_ Mar 18 2013