A223209 T(n,k)=Rolling icosahedron face footprints: number of nXk 0..19 arrays starting with 0 where 0..19 label faces of an icosahedron and every array movement to a horizontal or vertical neighbor moves across an icosahedral edge.

1, 3, 3, 9, 15, 9, 27, 75, 75, 27, 81, 375, 657, 375, 81, 243, 1875, 5763, 5763, 1875, 243, 729, 9375, 50553, 90111, 50553, 9375, 729, 2187, 46875, 443451, 1412907, 1412907, 443451, 46875, 2187, 6561, 234375, 3889953, 22163655, 39868737, 22163655

Offset: 1

Author

R. H. Hardin Mar 18 2013

Keywords

• nonn
• tabl

Comments

Table starts

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

......3........15...........75.............375...............1875

......9........75..........657............5763..............50553

.....27.......375.........5763...........90111............1412907

.....81......1875........50553.........1412907...........39868737

....243......9375.......443451........22163655.........1127761923

....729.....46875......3889953.......347696019........31921015497

...2187....234375.....34122675......5454600015.......903661481115

...6561...1171875....299324169.....85571052219.....25583075832465

..19683...5859375...2625672171...1342427863959....724276345970163

..59049..29296875..23032401201..21059839795875..20504869741550745

.177147.146484375.202040266467.330384125138847.580510427181846027

Examples

			Some solutions for n=3 k=4
..0..2..0..1....0..2..0..2....0..5..0..2....0..2..3..4....0..1..6.10
..2..8..2..0....2..8..2..8....5..9..5..0....2..0..2..3....1..6..1..6
..3..2..3..2....3..2..8.13....0..5..0..5....0..2..8..2....4..1..4..1
Face neighbors:
0 -> 1 2 5
1 -> 0 4 6
2 -> 0 3 8
3 -> 2 4 16
4 -> 3 1 17
5 -> 0 7 9
6 -> 1 7 10
7 -> 6 5 11
8 -> 2 9 13
9 -> 8 5 14
10 -> 6 12 17
11 -> 7 12 14
12 -> 11 10 19
13 -> 8 15 16
14 -> 9 11 15
15 -> 14 13 19
16 -> 3 13 18
17 -> 4 10 18
18 -> 16 17 19
19 -> 15 18 12
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..220

Crossrefs

Column 1 is A000244(n-1)

Column 2 is A005053

Formula

Empirical for column k:

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

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

k=3: a(n) = 9*a(n-1) -2*a(n-2)

k=4: a(n) = 19*a(n-1) -54*a(n-2) +32*a(n-3)

k=5: a(n) = 31*a(n-1) -24*a(n-2) -1612*a(n-3) +3816*a(n-4) +1152*a(n-5) -2784*a(n-6) +256*a(n-7)

k=6: [order 12]

k=7: [order 28]