|
%I #11 Aug 17 2018 09:22:07
%S 9,75,657,5763,50553,443451,3889953,34122675,299324169,2625672171,
%T 23032401201,202040266467,1772297595801,15546597829275,
%U 136374785271873,1196279871788307,10493769275551017,92051363736382539,807474735076340817
%N 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.
%C Column 3 of A223209.
%H R. H. Hardin, <a href="/A223204/b223204.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 9*a(n-1) - 2*a(n-2).
%F Conjectures from _Colin Barker_, Aug 17 2018: (Start)
%F G.f.: 3*x*(3 - 2*x) / (1 - 9*x + 2*x^2).
%F a(n) = (3*2^(-1-n)*((9-sqrt(73))^n*(3+sqrt(73)) + (-3+sqrt(73))*(9+sqrt(73))^n)) / sqrt(73).
%F (End)
%e Some solutions for n=3:
%e ..0..5..0....0..5..0....0..5..7....0..1..0....0..5..9....0..5..7....0..1..4
%e ..5..0..1....5..7..5....5..7..5....1..0..5....5..9..8....2..0..5....1..4.17
%e ..0..5..0....9..5..9....0..5..0....4..1..0....0..5..9....0..1..0....4.17.10
%e Face neighbors:
%e 0 -> 1 2 5
%e 1 -> 0 4 6
%e 2 -> 0 3 8
%e 3 -> 2 4 16
%e 4 -> 3 1 17
%e 5 -> 0 7 9
%e 6 -> 1 7 10
%e 7 -> 6 5 11
%e 8 -> 2 9 13
%e 9 -> 8 5 14
%e 10 -> 6 12 17
%e 11 -> 7 12 14
%e 12 -> 11 10 19
%e 13 -> 8 15 16
%e 14 -> 9 11 15
%e 15 -> 14 13 19
%e 16 -> 3 13 18
%e 17 -> 4 10 18
%e 18 -> 16 17 19
%e 19 -> 15 18 12
%Y Cf. A223209.
%K nonn
%O 1,1
%A _R. H. Hardin_, Mar 18 2013
| 