%I #4 Mar 19 2013 12:03:17

%S 1,3,8,9,27,64,27,189,243,512,81,1323,3969,2187,4096,243,9261,64827,

%T 83349,19683,32768,729,64827,1059723,3176523,1750329,177147,262144,

%U 2187,453789,17324685,121264857,155649627,36756909,1594323,2097152,6561

%N 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

%C Table starts

%C .........1..........3.............9................27....................81

%C .........8.........27...........189..............1323..................9261

%C ........64........243..........3969.............64827...............1059723

%C .......512.......2187.........83349...........3176523.............121264857

%C ......4096......19683.......1750329.........155649627...........13876429707

%C .....32768.....177147......36756909........7626831723.........1587890407761

%C ....262144....1594323.....771895089......373714754427.......181703507374179

%C ...2097152...14348907...16209796869....18312022966923.....20792470582897209

%C ..16777216..129140163..340405734249...897289125379227...2379298227030964827

%C .134217728.1162261467.7148520419229.43967167143582123.272264906211251105313

%C Horizontal or vertical instead of horizontal or antidiagonal gives A222444

%H R. H. Hardin, <a href="/A223331/b223331.txt">Table of n, a(n) for n = 1..199</a>

%F Empirical for column k:

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

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

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

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

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

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

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

%F Empirical for row n:

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

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

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

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

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

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

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

%e Some solutions for n=3 k=4

%e ..0..4..5..1....0..4..0..1....0..4..6..4....0..2..0..4....0..4..6..4

%e ..5..4..0..1....5..1..5..1....0..2..0..2....6..2..6..4....6..2..6..7

%e ..6..2..3..1....5..7..3..2....3..2..3..1....6..4..0..4....0..2..6..7

%e Vertex neighbors:

%e 0 -> 1 2 4

%e 1 -> 0 3 5

%e 2 -> 0 3 6

%e 3 -> 1 2 7

%e 4 -> 0 5 6

%e 5 -> 1 4 7

%e 6 -> 2 4 7

%e 7 -> 3 5 6

%Y Column 1 is A001018(n-1)

%Y Column 2 is A013708(n-1)

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

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

%Y Row 1 is A000244(n-1)

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

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_ Mar 19 2013