A223331 T(n,k)=Rolling cube footprints: number of nXk 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge

1, 3, 8, 9, 27, 64, 27, 189, 243, 512, 81, 1323, 3969, 2187, 4096, 243, 9261, 64827, 83349, 19683, 32768, 729, 64827, 1059723, 3176523, 1750329, 177147, 262144, 2187, 453789, 17324685, 121264857, 155649627, 36756909, 1594323, 2097152, 6561

Offset

1,2

Comments

Table starts

.........1..........3.............9................27....................81

.........8.........27...........189..............1323..................9261

........64........243..........3969.............64827...............1059723

.......512.......2187.........83349...........3176523.............121264857

......4096......19683.......1750329.........155649627...........13876429707

.....32768.....177147......36756909........7626831723.........1587890407761

....262144....1594323.....771895089......373714754427.......181703507374179

...2097152...14348907...16209796869....18312022966923.....20792470582897209

..16777216..129140163..340405734249...897289125379227...2379298227030964827

.134217728.1162261467.7148520419229.43967167143582123.272264906211251105313

Horizontal or vertical instead of horizontal or antidiagonal gives A222444

Links

R. H. Hardin, Table of n, a(n) for n = 1..199

Formula

Empirical for column k:

k=1: a(n) = 8*a(n-1)

k=2: a(n) = 9*a(n-1)

k=3: a(n) = 21*a(n-1)

k=4: a(n) = 49*a(n-1)

k=5: a(n) = 117*a(n-1) -294*a(n-2)

k=6: a(n) = 282*a(n-1) -3969*a(n-2) +9604*a(n-3)

k=7: a(n) = 692*a(n-1) -43569*a(n-2) +847042*a(n-3) -6303164*a(n-4) +15731352*a(n-5)

Empirical for row n:

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

n=2: a(n) = 7*a(n-1) for n>2

n=3: a(n) = 18*a(n-1) -27*a(n-2) for n>4

n=4: a(n) = 48*a(n-1) -402*a(n-2) +1064*a(n-3) -789*a(n-4) for n>7

n=5: [order 9] for n>13

n=6: [order 20] for n>25

n=7: [order 51] for n>57

Example

Some solutions for n=3 k=4
..0..4..5..1....0..4..0..1....0..4..6..4....0..2..0..4....0..4..6..4
..5..4..0..1....5..1..5..1....0..2..0..2....6..2..6..4....6..2..6..7
..6..2..3..1....5..7..3..2....3..2..3..1....6..4..0..4....0..2..6..7
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

Crossrefs

Column 1 is A001018(n-1)

Column 2 is A013708(n-1)

Column 3 is 9*21^(n-1)

Column 4 is 27*49^(n-1)

Row 1 is A000244(n-1)

Row 2 is 27*7^(n-2) for n>1

Sequence in context: A167344 A025615 A297324 * A101720 A093439 A000927

Adjacent sequences: A223328 A223329 A223330 * A223332 A223333 A223334

Keyword

nonn,tabl

Author

R. H. Hardin Mar 19 2013

Status

approved