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A223228 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.

%I A223228 #10 Aug 17 2018 09:22:28
%S A223228 25,785,25225,812225,26157625,842416625,27130395625,873746350625,
%T A223228 28139386665625,906241361740625,29185902861015625,939944877578890625,
%U A223228 30271339457769765625,974901842039841640625,31397143920195178515625
%N A223228 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 A223228 Column 3 of A223233.
%H A223228 R. H. Hardin, <a href="/A223228/b223228.txt">Table of n, a(n) for n = 1..210</a>
%F A223228 Empirical: a(n) = 35*a(n-1) - 90*a(n-2).
%F A223228 Conjectures from _Colin Barker_, Aug 17 2018: (Start)
%F A223228 G.f.: 5*x*(5 - 18*x) / (1 - 35*x + 90*x^2).
%F A223228 a(n) = (2^(-1-n)*((35-sqrt(865))^n*(-15+sqrt(865)) + (15+sqrt(865))*(35+sqrt(865))^n)) / sqrt(865).
%F A223228 (End)
%e A223228 Some solutions for n=3:
%e A223228 ..0..7.11....0..7.11....0..6.10....0..7..0....0..2..0....0..2..0....0..6..4
%e A223228 ..3..7..3....3..7..1....2..6..2...11..7..1....0..7..0....0..6..4....4..2..4
%e A223228 .11..7.11....5..7..3....2..4..2....0..7..5...11..7..3....4..2..0....8..9..4
%e A223228 Vertex neighbors:
%e A223228 0 -> 1 2 5 6 7
%e A223228 1 -> 0 2 3 7 8
%e A223228 2 -> 0 1 4 6 8
%e A223228 3 -> 1 7 8 9 11
%e A223228 4 -> 2 6 8 9 10
%e A223228 5 -> 0 6 7 10 11
%e A223228 6 -> 0 2 4 5 10
%e A223228 7 -> 0 1 3 5 11
%e A223228 8 -> 1 2 3 4 9
%e A223228 9 -> 3 4 8 10 11
%e A223228 10 -> 4 5 6 9 11
%e A223228 11 -> 3 5 7 9 10
%Y A223228 Cf. A223233.
%K A223228 nonn
%O A223228 1,1
%A A223228 _R. H. Hardin_, Mar 18 2013