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
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
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
Comments