A223209 -id:A223209 - cp's OEIS frontend

A223204 Rolling icosahedron face footprints: number of n X 3 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.

9, 75, 657, 5763, 50553, 443451, 3889953, 34122675, 299324169, 2625672171, 23032401201, 202040266467, 1772297595801, 15546597829275, 136374785271873, 1196279871788307, 10493769275551017, 92051363736382539, 807474735076340817

Offset: 1

R. H. Hardin, Mar 18 2013

nonn

Column 3 of A223209.

			Some solutions for n=3:
..0..5..0....0..5..0....0..5..7....0..1..0....0..5..9....0..5..7....0..1..4
..5..0..1....5..7..5....5..7..5....1..0..5....5..9..8....2..0..5....1..4.17
..0..5..0....9..5..9....0..5..0....4..1..0....0..5..9....0..1..0....4.17.10
Face neighbors:
0 -> 1 2 5
1 -> 0 4 6
2 -> 0 3 8
3 -> 2 4 16
4 -> 3 1 17
5 -> 0 7 9
6 -> 1 7 10
7 -> 6 5 11
8 -> 2 9 13
9 -> 8 5 14
10 -> 6 12 17
11 -> 7 12 14
12 -> 11 10 19
13 -> 8 15 16
14 -> 9 11 15
15 -> 14 13 19
16 -> 3 13 18
17 -> 4 10 18
18 -> 16 17 19
19 -> 15 18 12

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A223209.

Empirical: a(n) = 9*a(n-1) - 2*a(n-2).
Conjectures from Colin Barker, Aug 17 2018: (Start)
G.f.: 3*x*(3 - 2*x) / (1 - 9*x + 2*x^2).
a(n) = (3*2^(-1-n)*((9-sqrt(73))^n*(3+sqrt(73)) + (-3+sqrt(73))*(9+sqrt(73))^n)) / sqrt(73).
(End)

A223205 Rolling icosahedron face footprints: number of n X 4 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.

Original entry on oeis.org

27, 375, 5763, 90111, 1412907, 22163655, 347696019, 5454600015, 85571052219, 1342427863959, 21059839795875, 330384125138847, 5183024720307531, 81310641801813351, 1275591151342290099, 20011314009255431919

Offset: 1

R. H. Hardin, Mar 18 2013

nonn

Column 4 of A223209.

			Some solutions for n=3:
..0..1..6.10....0..2..3..2....0..5..9..8....0..1..0..2....0..1..4.17
..1..0..1..6....2..3..2..3....5..9.14..9....1..6..1..0....1..0..1..4
..4..1..6.10....3..2..3.16....9.14..9..5....0..1..0..2....4..1..6..1
Face neighbors:
0 -> 1 2 5
1 -> 0 4 6
2 -> 0 3 8
3 -> 2 4 16
4 -> 3 1 17
5 -> 0 7 9
6 -> 1 7 10
7 -> 6 5 11
8 -> 2 9 13
9 -> 8 5 14
10 -> 6 12 17
11 -> 7 12 14
12 -> 11 10 19
13 -> 8 15 16
14 -> 9 11 15
15 -> 14 13 19
16 -> 3 13 18
17 -> 4 10 18
18 -> 16 17 19
19 -> 15 18 12

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A223209.

Empirical: a(n) = 19*a(n-1) - 54*a(n-2) + 32*a(n-3).

Empirical g.f.: 3*x*(9 - 46*x + 32*x^2) / (1 - 19*x + 54*x^2 - 32*x^3). - Colin Barker, Aug 17 2018

A223206 Rolling icosahedron face footprints: number of nX5 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.

Original entry on oeis.org

81, 1875, 50553, 1412907, 39868737, 1127761923, 31921015497, 903661481115, 25583075832465, 724276345970163, 20504869741550745, 580510427181846027, 16434750355138945761, 465281963351360897763, 13172534004090254190441

Offset: 1

R. H. Hardin Mar 18 2013

nonn

Column 5 of A223209

			Some solutions for n=3
..0..1..4..3..4....0..2..0..2..0....0..1..0..2..3....0..2..0..1..6
..1..4..3..4..1....2..8..2..0..2....5..0..5..0..2....1..0..1..4..1
..4..1..4..1..6....8..2..3..2..3....0..2..0..5..0....0..1..4..1..4
Face neighbors:
0 -> 1 2 5
1 -> 0 4 6
2 -> 0 3 8
3 -> 2 4 16
4 -> 3 1 17
5 -> 0 7 9
6 -> 1 7 10
7 -> 6 5 11
8 -> 2 9 13
9 -> 8 5 14
10 -> 6 12 17
11 -> 7 12 14
12 -> 11 10 19
13 -> 8 15 16
14 -> 9 11 15
15 -> 14 13 19
16 -> 3 13 18
17 -> 4 10 18
18 -> 16 17 19
19 -> 15 18 12

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: 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)

A223207 Rolling icosahedron face footprints: number of nX6 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.

Original entry on oeis.org

243, 9375, 443451, 22163655, 1127761923, 57728430255, 2960941376139, 151970441720919, 7801619707803603, 400536865551609279, 20564155823714361819, 1055802848340690163431, 54207072655600127312547

Offset: 1

R. H. Hardin Mar 18 2013

nonn

Column 6 of A223209

			Some solutions for n=3
..0..2..0..1..6.10....0..1..6.10..6.10....0..5..0..2..3..4....0..5..9..5..9..8
..5..0..1..0..1..6....5..0..1..6..1..6....5..0..2..3.16..3....5..9..5..9..8..9
..0..2..0..1..6.10....0..1..4..1..6..1....0..2..3..2..3.16....0..5..0..5..9.14
Face neighbors:
0 -> 1 2 5
1 -> 0 4 6
2 -> 0 3 8
3 -> 2 4 16
4 -> 3 1 17
5 -> 0 7 9
6 -> 1 7 10
7 -> 6 5 11
8 -> 2 9 13
9 -> 8 5 14
10 -> 6 12 17
11 -> 7 12 14
12 -> 11 10 19
13 -> 8 15 16
14 -> 9 11 15
15 -> 14 13 19
16 -> 3 13 18
17 -> 4 10 18
18 -> 16 17 19
19 -> 15 18 12

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 77*a(n-1) -1444*a(n-2) +5860*a(n-3) +38744*a(n-4) -300016*a(n-5) +339296*a(n-6) +1286720*a(n-7) -2655360*a(n-8) -56832*a(n-9) +1682944*a(n-10) -149504*a(n-11) -163840*a(n-12)

A223208 Rolling icosahedron face footprints: number of nX7 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.

Original entry on oeis.org

729, 46875, 3889953, 347696019, 31921015497, 2960941376139, 275769851453745, 25725360515161923, 2401334251012194777, 224209069529029889211, 20936162679299127225537, 1955051227721359130017011

Offset: 1

R. H. Hardin Mar 18 2013

nonn

Column 7 of A223209

			Some solutions for n=3
..0..5..9..5..9..5..7....0..5..0..2..0..1..4....0..5..0..1..6..1..4
..5..0..5..0..5..9..5....5..0..2..0..5..0..1....5..0..1..0..1..4.17
..0..5..0..1..0..5..7....0..5..0..5..0..2..0....0..5..0..1..4.17..4
Face neighbors:
0 -> 1 2 5
1 -> 0 4 6
2 -> 0 3 8
3 -> 2 4 16
4 -> 3 1 17
5 -> 0 7 9
6 -> 1 7 10
7 -> 6 5 11
8 -> 2 9 13
9 -> 8 5 14
10 -> 6 12 17
11 -> 7 12 14
12 -> 11 10 19
13 -> 8 15 16
14 -> 9 11 15
15 -> 14 13 19
16 -> 3 13 18
17 -> 4 10 18
18 -> 16 17 19
19 -> 15 18 12

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 117*a(n-1) -614*a(n-2) -193608*a(n-3) +4171896*a(n-4) +11415328*a(n-5) -842180224*a(n-6) +3384845504*a(n-7) +50528534400*a(n-8) -356768428544*a(n-9) -786566761984*a(n-10) +10681051645952*a(n-11) -6785526038528*a(n-12) -118510105321472*a(n-13) +197530250887168*a(n-14) +559297782349824*a(n-15) -1309510355714048*a(n-16) -1087672042455040*a(n-17) +3564507406925824*a(n-18) +719985279238144*a(n-19) -4461650060509184*a(n-20) +91947477762048*a(n-21) +2549613680656384*a(n-22) -206006306996224*a(n-23) -619282299879424*a(n-24) +65421579386880*a(n-25) +47138239152128*a(n-26) -8245531901952*a(n-27) +260919263232*a(n-28)

A223203 Rolling icosahedron face footprints: number of n X n 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.

Original entry on oeis.org

1, 15, 657, 90111, 39868737, 57728430255, 275769851453745, 4367470887097715487, 230051556728875753924449, 40390820217552511124819066703

Offset: 1

R. H. Hardin Mar 18 2013

nonn

Diagonal of A223209

			Some solutions for n=3
..0..2..0....0..5..7....0..5..0....0..5..0....0..5..7....0..2..0....0..1..6
..1..0..2....5..0..5....2..0..5....5..0..5....5..7.11....2..3..2....1..0..1
..0..5..0....0..5..9....0..2..0....0..5..7....7.11.12....8..2..3....6..1..6
Face neighbors:
0 -> 1 2 5
1 -> 0 4 6
2 -> 0 3 8
3 -> 2 4 16
4 -> 3 1 17
5 -> 0 7 9
6 -> 1 7 10
7 -> 6 5 11
8 -> 2 9 13
9 -> 8 5 14
10 -> 6 12 17
11 -> 7 12 14
12 -> 11 10 19
13 -> 8 15 16
14 -> 9 11 15
15 -> 14 13 19
16 -> 3 13 18
17 -> 4 10 18
18 -> 16 17 19
19 -> 15 18 12

cp's OEIS Frontend

A223204 Rolling icosahedron face footprints: number of n X 3 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.

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A223205 Rolling icosahedron face footprints: number of n X 4 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.

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A223206 Rolling icosahedron face footprints: number of nX5 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.

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A223207 Rolling icosahedron face footprints: number of nX6 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.

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A223208 Rolling icosahedron face footprints: number of nX7 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.

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Examples

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Formula

A223203 Rolling icosahedron face footprints: number of n X n 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.

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