cp's OEIS Frontend

A223321 T(n,k)=Rolling icosahedron footprints: number of nXk 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal or antidiagonal neighbor moves along an icosahedral edge.

%I A223321 #6 Jul 23 2025 04:00:22
%S A223321 1,5,12,25,125,144,125,1625,3125,1728,625,21125,105625,78125,20736,
%T A223321 3125,274625,3570125,6865625,1953125,248832,15625,3570125,122039125,
%U A223321 603351125,446265625,48828125,2985984,78125,46411625,4176940625,54279694625
%N A223321 T(n,k)=Rolling icosahedron footprints: number of nXk 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal or antidiagonal neighbor moves along an icosahedral edge.
%C A223321 Table starts
%C A223321 ........1...........5..............25................125....................625
%C A223321 .......12.........125............1625..............21125.................274625
%C A223321 ......144........3125..........105625............3570125..............122039125
%C A223321 .....1728.......78125.........6865625..........603351125............54279694625
%C A223321 ....20736.....1953125.......446265625.......101966340125.........24143758634125
%C A223321 ...248832....48828125.....29007265625.....17232311481125......10739266230499625
%C A223321 ..2985984..1220703125...1885472265625...2912260640310125....4776881955584279125
%C A223321 .35831808.30517578125.122555697265625.492172048212411125.2124782217358970404625
%H A223321 R. H. Hardin, <a href="/A223321/b223321.txt">Table of n, a(n) for n = 1..97</a>
%F A223321 Empirical for column k:
%F A223321 k=1: a(n) = 12*a(n-1)
%F A223321 k=2: a(n) = 25*a(n-1)
%F A223321 k=3: a(n) = 65*a(n-1)
%F A223321 k=4: a(n) = 169*a(n-1)
%F A223321 k=5: a(n) = 479*a(n-1) -15210*a(n-2)
%F A223321 k=6: a(n) = 1366*a(n-1) -232713*a(n-2) +9253764*a(n-3)
%F A223321 k=7: [order 8]
%F A223321 Empirical for row n:
%F A223321 n=1: a(n) = 5*a(n-1)
%F A223321 n=2: a(n) = 13*a(n-1) for n>2
%F A223321 n=3: a(n) = 38*a(n-1) -129*a(n-2) for n>4
%F A223321 n=4: [order 7] for n>10
%F A223321 n=5: [order 32] for n>36
%e A223321 Some solutions for n=3 k=4
%e A223321 ..0..1..8..9....0..1..0..7....0..1..0..2....0..1..0..6....0..6..2..4
%e A223321 ..0..2..8..2....0..5..0..5....0..6..0..2....0..6.10..5....0..1..2..4
%e A223321 ..6..2..4..2....0..1..0..7....0..7..0..7....0..6.10..5....0..6.10..4
%e A223321 Vertex neighbors:
%e A223321 0 -> 1 2 5 6 7
%e A223321 1 -> 0 2 3 7 8
%e A223321 2 -> 0 1 4 6 8
%e A223321 3 -> 1 7 8 9 11
%e A223321 4 -> 2 6 8 9 10
%e A223321 5 -> 0 6 7 10 11
%e A223321 6 -> 0 2 4 5 10
%e A223321 7 -> 0 1 3 5 11
%e A223321 8 -> 1 2 3 4 9
%e A223321 9 -> 3 4 8 10 11
%e A223321 10 -> 4 5 6 9 11
%e A223321 11 -> 3 5 7 9 10
%Y A223321 Column 1 is A001021(n-1)
%Y A223321 Column 2 is A013710(n-1)
%Y A223321 Column 3 is 25*65^(n-1)
%Y A223321 Column 4 is 125*169^(n-1)
%Y A223321 Row 1 is A000351(n-1)
%K A223321 nonn,tabl
%O A223321 1,2
%A A223321 _R. H. Hardin_ Mar 19 2013