|
%I #8 Aug 17 2018 09:21:46
%S 25,80,400,2080,10880,56960,298240,1561600,8176640,42813440,224174080,
%T 1173790720,6146048000,32181125120,168502558720,882290851840,
%U 4619734876160,24189245849600,126656535592960,663182230159360
%N Rolling icosahedron footprints: number of n X 3 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.
%C Column 3 of A223186.
%H R. H. Hardin, <a href="/A223181/b223181.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 6*a(n-1) - 4*a(n-2) for n>3.
%F Conjectures from _Colin Barker_, Aug 17 2018: (Start)
%F G.f.: 5*x*(5 - 14*x + 4*x^2) / (1 - 6*x + 4*x^2).
%F a(n) = sqrt(5)*((3-sqrt(5))^n*(1+sqrt(5)) + (-1+sqrt(5))*(3+sqrt(5))^n) for n>1.
%F (End)
%e Some solutions for n=3:
%e ..0..7..1....0..2..6....0..5..7....0..2..6....0..5.10....0..6..4....0..7..1
%e ..5..0..2....6..0..2....7.11..5....6..4..2....7.11..5....5.10..6....1..0..2
%e ..7..1..0....2..1..0....5..7.11....2..6..4....5.10..6....6..5..0....2..1..8
%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 Cf. A223186.
%K nonn
%O 1,1
%A _R. H. Hardin_, Mar 18 2013
| 