A223204 Rolling icosahedron face footprints: number of n X 3 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.
9, 75, 657, 5763, 50553, 443451, 3889953, 34122675, 299324169, 2625672171, 23032401201, 202040266467, 1772297595801, 15546597829275, 136374785271873, 1196279871788307, 10493769275551017, 92051363736382539, 807474735076340817
Offset: 1
Keywords
Examples
Some solutions for n=3: ..0..5..0....0..5..0....0..5..7....0..1..0....0..5..9....0..5..7....0..1..4 ..5..0..1....5..7..5....5..7..5....1..0..5....5..9..8....2..0..5....1..4.17 ..0..5..0....9..5..9....0..5..0....4..1..0....0..5..9....0..1..0....4.17.10 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..210
Crossrefs
Cf. A223209.
Formula
Empirical: a(n) = 9*a(n-1) - 2*a(n-2).
Conjectures from Colin Barker, Aug 17 2018: (Start)
G.f.: 3*x*(3 - 2*x) / (1 - 9*x + 2*x^2).
a(n) = (3*2^(-1-n)*((9-sqrt(73))^n*(3+sqrt(73)) + (-3+sqrt(73))*(9+sqrt(73))^n)) / sqrt(73).
(End)
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