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.

1, 4, 6, 16, 48, 36, 64, 576, 576, 216, 256, 6144, 20992, 6912, 1296, 1024, 67584, 622592, 765952, 82944, 7776, 4096, 737280, 19726336, 63438848, 27951104, 995328, 46656, 16384, 8060928, 611319808, 5889851392, 6467616768, 1020002304, 11943936

Offset: 1

Author

R. H. Hardin Mar 19 2013

Keywords

• nonn
• tabl

Comments

Table starts

....1......4.........16...........64.............256...............1024

....6.....48........576.........6144...........67584.............737280

...36....576......20992.......622592........19726336..........611319808

..216...6912.....765952.....63438848......5889851392.......522106961920

.1296..82944...27951104...6467616768...1771674009600....450204914417664

.7776.995328.1020002304.659411697664.534392715870208.389343801904201728

Examples

			Some solutions for n=3 k=4
..0..3..1..2....0..1..0..1....0..4..5..1....0..4..2..4....0..2..1..3
..0..2..4..3....0..3..5..1....0..4..0..3....0..1..0..4....0..3..4..2
..4..2..1..2....0..2..0..1....3..1..5..4....3..4..0..1....0..3..4..0
Face neighbors:
0.->.1.2.3.4
1.->.0.2.3.5
2.->.0.1.4.5
3.->.0.1.4.5
4.->.0.3.2.5
5.->.1.3.4.2
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..311

Crossrefs

Column 1 is A000400(n-1)

Column 2 is 4*12^(n-1)

Column 3 is A223197

Row 1 is A000302(n-1)

Formula

Empirical for column k:

k=1: a(n) = 6*a(n-1)

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

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

k=4: a(n) = 112*a(n-1) -1024*a(n-2)

k=5: [order 6]

k=6: [order 9]

k=7: [order 19]

Empirical for row n:

n=1: a(n) = 4*a(n-1)

n=2: a(n) = 8*a(n-1) +32*a(n-2)

n=3: a(n) = 24*a(n-1) +256*a(n-2) -1024*a(n-3) for n>4

n=4: [order 6] for n>7

n=5: [order 10] for n>11

n=6: [order 23] for n>24