A223321 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.

1, 5, 12, 25, 125, 144, 125, 1625, 3125, 1728, 625, 21125, 105625, 78125, 20736, 3125, 274625, 3570125, 6865625, 1953125, 248832, 15625, 3570125, 122039125, 603351125, 446265625, 48828125, 2985984, 78125, 46411625, 4176940625, 54279694625

Offset: 1

Author

R. H. Hardin Mar 19 2013

Keywords

• nonn
• tabl

Comments

Table starts

........1...........5..............25................125....................625

.......12.........125............1625..............21125.................274625

......144........3125..........105625............3570125..............122039125

.....1728.......78125.........6865625..........603351125............54279694625

....20736.....1953125.......446265625.......101966340125.........24143758634125

...248832....48828125.....29007265625.....17232311481125......10739266230499625

..2985984..1220703125...1885472265625...2912260640310125....4776881955584279125

.35831808.30517578125.122555697265625.492172048212411125.2124782217358970404625

Examples

			Some solutions for n=3 k=4
..0..1..8..9....0..1..0..7....0..1..0..2....0..1..0..6....0..6..2..4
..0..2..8..2....0..5..0..5....0..6..0..2....0..6.10..5....0..1..2..4
..6..2..4..2....0..1..0..7....0..7..0..7....0..6.10..5....0..6.10..4
Vertex neighbors:
0 -> 1 2 5 6 7
1 -> 0 2 3 7 8
2 -> 0 1 4 6 8
3 -> 1 7 8 9 11
4 -> 2 6 8 9 10
5 -> 0 6 7 10 11
6 -> 0 2 4 5 10
7 -> 0 1 3 5 11
8 -> 1 2 3 4 9
9 -> 3 4 8 10 11
10 -> 4 5 6 9 11
11 -> 3 5 7 9 10
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..97

Crossrefs

Column 1 is A001021(n-1)

Column 2 is A013710(n-1)

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

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

Row 1 is A000351(n-1)

Formula

Empirical for column k:

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

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

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

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

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

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

k=7: [order 8]

Empirical for row n:

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

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

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

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

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