cp's OEIS Frontend

A223209 T(n,k)=Rolling icosahedron face footprints: number of nXk 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.

%I A223209 #6 Jul 23 2025 03:51:53
%S A223209 1,3,3,9,15,9,27,75,75,27,81,375,657,375,81,243,1875,5763,5763,1875,
%T A223209 243,729,9375,50553,90111,50553,9375,729,2187,46875,443451,1412907,
%U A223209 1412907,443451,46875,2187,6561,234375,3889953,22163655,39868737,22163655
%N A223209 T(n,k)=Rolling icosahedron face footprints: number of nXk 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 A223209 Table starts
%C A223209 ......1.........3............9..............27.................81
%C A223209 ......3........15...........75.............375...............1875
%C A223209 ......9........75..........657............5763..............50553
%C A223209 .....27.......375.........5763...........90111............1412907
%C A223209 .....81......1875........50553.........1412907...........39868737
%C A223209 ....243......9375.......443451........22163655.........1127761923
%C A223209 ....729.....46875......3889953.......347696019........31921015497
%C A223209 ...2187....234375.....34122675......5454600015.......903661481115
%C A223209 ...6561...1171875....299324169.....85571052219.....25583075832465
%C A223209 ..19683...5859375...2625672171...1342427863959....724276345970163
%C A223209 ..59049..29296875..23032401201..21059839795875..20504869741550745
%C A223209 .177147.146484375.202040266467.330384125138847.580510427181846027
%H A223209 R. H. Hardin, <a href="/A223209/b223209.txt">Table of n, a(n) for n = 1..220</a>
%F A223209 Empirical for column k:
%F A223209 k=1: a(n) = 3*a(n-1)
%F A223209 k=2: a(n) = 5*a(n-1)
%F A223209 k=3: a(n) = 9*a(n-1) -2*a(n-2)
%F A223209 k=4: a(n) = 19*a(n-1) -54*a(n-2) +32*a(n-3)
%F A223209 k=5: a(n) = 31*a(n-1) -24*a(n-2) -1612*a(n-3) +3816*a(n-4) +1152*a(n-5) -2784*a(n-6) +256*a(n-7)
%F A223209 k=6: [order 12]
%F A223209 k=7: [order 28]
%e A223209 Some solutions for n=3 k=4
%e A223209 ..0..2..0..1....0..2..0..2....0..5..0..2....0..2..3..4....0..1..6.10
%e A223209 ..2..8..2..0....2..8..2..8....5..9..5..0....2..0..2..3....1..6..1..6
%e A223209 ..3..2..3..2....3..2..8.13....0..5..0..5....0..2..8..2....4..1..4..1
%e A223209 Face neighbors:
%e A223209 0 -> 1 2 5
%e A223209 1 -> 0 4 6
%e A223209 2 -> 0 3 8
%e A223209 3 -> 2 4 16
%e A223209 4 -> 3 1 17
%e A223209 5 -> 0 7 9
%e A223209 6 -> 1 7 10
%e A223209 7 -> 6 5 11
%e A223209 8 -> 2 9 13
%e A223209 9 -> 8 5 14
%e A223209 10 -> 6 12 17
%e A223209 11 -> 7 12 14
%e A223209 12 -> 11 10 19
%e A223209 13 -> 8 15 16
%e A223209 14 -> 9 11 15
%e A223209 15 -> 14 13 19
%e A223209 16 -> 3 13 18
%e A223209 17 -> 4 10 18
%e A223209 18 -> 16 17 19
%e A223209 19 -> 15 18 12
%Y A223209 Column 1 is A000244(n-1)
%Y A223209 Column 2 is A005053
%K A223209 nonn,tabl
%O A223209 1,2
%A A223209 _R. H. Hardin_ Mar 18 2013