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

Author

R. H. Hardin Mar 19 2013

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

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

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) [phone]*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