A223321 - OEIS

%I #4 Mar 19 2013 11:30:20

%S 1,5,12,25,125,144,125,1625,3125,1728,625,21125,105625,78125,20736,

%T 3125,274625,3570125,6865625,1953125,248832,15625,3570125,122039125,

%U 603351125,446265625,48828125,2985984,78125,46411625,4176940625,54279694625

%N T(n,k)=Rolling icosahedron footprints: number of nXk 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal or antidiagonal neighbor moves along an icosahedral edge

%C Table starts

%C ........1...........5..............25................125....................625

%C .......12.........125............1625..............21125.................274625

%C ......144........3125..........105625............3570125..............122039125

%C .....1728.......78125.........6865625..........603351125............54279694625

%C ....20736.....1953125.......446265625.......101966340125.........24143758634125

%C ...248832....48828125.....29007265625.....17232311481125......10739266230499625

%C ..2985984..1220703125...1885472265625...2912260640310125....4776881955584279125

%C .35831808.30517578125.122555697265625.492172048212411125.2124782217358970404625

%H R. H. Hardin, <a href="/A223321/b223321.txt">Table of n, a(n) for n = 1..97</a>

%F Empirical for column k:

%F k=1: a(n) = 12*a(n-1)

%F k=2: a(n) = 25*a(n-1)

%F k=3: a(n) = 65*a(n-1)

%F k=4: a(n) = 169*a(n-1)

%F k=5: a(n) = 479*a(n-1) -15210*a(n-2)

%F k=6: a(n) = 1366*a(n-1) -232713*a(n-2) +9253764*a(n-3)

%F k=7: [order 8]

%F Empirical for row n:

%F n=1: a(n) = 5*a(n-1)

%F n=2: a(n) = 13*a(n-1) for n>2

%F n=3: a(n) = 38*a(n-1) -129*a(n-2) for n>4

%F n=4: [order 7] for n>10

%F n=5: [order 32] for n>36

%e Some solutions for n=3 k=4

%e ..0..1..8..9....0..1..0..7....0..1..0..2....0..1..0..6....0..6..2..4

%e ..0..2..8..2....0..5..0..5....0..6..0..2....0..6.10..5....0..1..2..4

%e ..6..2..4..2....0..1..0..7....0..7..0..7....0..6.10..5....0..6.10..4

%e Vertex neighbors:

%e 0 -> 1 2 5 6 7

%e 1 -> 0 2 3 7 8

%e 2 -> 0 1 4 6 8

%e 3 -> 1 7 8 9 11

%e 4 -> 2 6 8 9 10

%e 5 -> 0 6 7 10 11

%e 6 -> 0 2 4 5 10

%e 7 -> 0 1 3 5 11

%e 8 -> 1 2 3 4 9

%e 9 -> 3 4 8 10 11

%e 10 -> 4 5 6 9 11

%e 11 -> 3 5 7 9 10

%Y Column 1 is A001021(n-1)

%Y Column 2 is A013710(n-1)

%Y Column 3 is 25*65^(n-1)

%Y Column 4 is 125*169^(n-1)

%Y Row 1 is A000351(n-1)

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_ Mar 19 2013