A223228 - OEIS

%I #10 Aug 17 2018 09:22:28

%S 25,785,25225,812225,26157625,842416625,27130395625,873746350625,

%T 28139386665625,906241361740625,29185902861015625,939944877578890625,

%U 30271339457769765625,974901842039841640625,31397143920195178515625

%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, diagonal or antidiagonal neighbor moves along an icosahedral edge.

%C Column 3 of A223233.

%H R. H. Hardin, <a href="/A223228/b223228.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 35*a(n-1) - 90*a(n-2).

%F Conjectures from _Colin Barker_, Aug 17 2018: (Start)

%F G.f.: 5*x*(5 - 18*x) / (1 - 35*x + 90*x^2).

%F a(n) = (2^(-1-n)*((35-sqrt(865))^n*(-15+sqrt(865)) + (15+sqrt(865))*(35+sqrt(865))^n)) / sqrt(865).

%F (End)

%e Some solutions for n=3:

%e ..0..7.11....0..7.11....0..6.10....0..7..0....0..2..0....0..2..0....0..6..4

%e ..3..7..3....3..7..1....2..6..2...11..7..1....0..7..0....0..6..4....4..2..4

%e .11..7.11....5..7..3....2..4..2....0..7..5...11..7..3....4..2..0....8..9..4

%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. A223233.

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 18 2013