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A223202 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.

%I A223202 #6 Jul 23 2025 03:51:18
%S A223202 1,4,4,16,48,16,64,576,576,64,256,6912,20992,6912,256,1024,82944,
%T A223202 765952,765952,82944,1024,4096,995328,27951104,85327872,27951104,
%U A223202 995328,4096,16384,11943936,1020002304,9515827200,9515827200,1020002304,11943936
%N A223202 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 A223202 Table starts
%C A223202 ....1......4.........16............64..............256................1024
%C A223202 ....4.....48........576..........6912............82944..............995328
%C A223202 ...16....576......20992........765952.........27951104..........1020002304
%C A223202 ...64...6912.....765952......85327872.......9515827200.......1061444124672
%C A223202 ..256..82944...27951104....9515827200....3249109204992....1110327429169152
%C A223202 .1024.995328.1020002304.1061444124672.1110327429169152.1163614255186968576
%H A223202 R. H. Hardin, <a href="/A223202/b223202.txt">Table of n, a(n) for n = 1..364</a>
%F A223202 Empirical for column k:
%F A223202 k=1: a(n) = 4*a(n-1)
%F A223202 k=2: a(n) = 12*a(n-1)
%F A223202 k=3: a(n) = 40*a(n-1) -128*a(n-2)
%F A223202 k=4: a(n) = 144*a(n-1) -3840*a(n-2) +24576*a(n-3)
%F A223202 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 A223202 k=6: [order 7]
%F A223202 k=7: [order 13]
%e A223202 Some solutions for n=3 k=4
%e A223202 ..0..2..4..2....0..1..5..4....0..3..4..2....0..2..0..4....0..2..0..3
%e A223202 ..3..1..2..5....3..5..3..0....3..1..3..5....3..5..2..0....3..5..2..5
%e A223202 ..0..3..1..2....0..2..0..2....0..3..5..4....1..3..5..2....1..2..1..2
%e A223202 Face neighbors:
%e A223202 0,5 -> 1 2 3 4
%e A223202 1,4 -> 0 2 3 5
%e A223202 2,3 -> 0 1 4 5
%Y A223202 Column 1 is A000302(n-1)
%Y A223202 Column 2 is 4*12^(k-1)
%K A223202 nonn,tabl
%O A223202 1,2
%A A223202 _R. H. Hardin_ Mar 18 2013