A223357 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 antidiagonal neighbor moves across a corresponding cube edge

1, 4, 6, 16, 64, 36, 64, 768, 1024, 216, 256, 9216, 36864, 16384, 1296, 1024, 110592, 1327104, 1769472, 262144, 7776, 4096, 1327104, 48365568, 191102976, 84934656, 4194304, 46656, 16384, 15925248, 1764753408, 21177040896, 27518828544

Offset

1,2

Comments

Table starts

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

....6......64........768..........9216...........110592.............1327104

...36....1024......36864.......1327104.........48365568..........1764753408

..216...16384....1769472.....191102976......21177040896.......2356125106176

.1296..262144...84934656...27518828544....9273505480704....3147420753985536

.7776.4194304.4076863488.3962711310336.4060947412942848.4204783428144463872

Links

R. H. Hardin, Table of n, a(n) for n = 1..241

Formula

Empirical for column k:

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

k=2: a(n) = 16*a(n-1)

k=3: a(n) = 48*a(n-1)

k=4: a(n) = 144*a(n-1)

k=5: a(n) = 480*a(n-1) -18432*a(n-2)

k=6: a(n) = 1600*a(n-1) -368640*a(n-2) +21233664*a(n-3)

k=7: a(n) = 5376*a(n-1) -5750784*a(n-2) +2038431744*a(n-3) -217432719360*a(n-4)

Empirical for row n:

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

n=2: a(n) = 12*a(n-1) for n>2

n=3: a(n) = 40*a(n-1) -128*a(n-2) for n>4

n=4: a(n) = 144*a(n-1) -3840*a(n-2) +24576*a(n-3) for n>7

n=5: [order 7] for n>11

n=6: [order 9] for n>15

n=7: [order 27] for n>33

Example

Some solutions for n=3 k=4
..0..2..0..2....0..3..5..2....0..2..5..4....0..4..0..4....0..1..3..4
..0..4..0..1....0..3..5..3....0..4..0..1....0..4..0..4....0..1..0..1
..0..4..5..4....0..1..0..1....0..4..2..1....0..2..0..1....0..1..2..4

Crossrefs

Column 1 is A000400(n-1)

Column 2 is A013709 (n-1)

Column 3 is 16*48^(n-1)

Column 4 is 64*144^(n-1)

Row 1 is A000302(n-1)

Row 2 is 64*12^(n-2) for n>1

Sequence in context: A223269 A264471 A264477 * A083009 A365111 A190968

Adjacent sequences: A223354 A223355 A223356 * A223358 A223359 A223360

Keyword

nonn,tabl

Author

R. H. Hardin Mar 19 2013

Status

approved