A223186 -id:A223186 - cp's OEIS frontend

A223181 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, vertical or antidiagonal neighbor moves along an icosahedral edge.

25, 80, 400, 2080, 10880, 56960, 298240, 1561600, 8176640, 42813440, 224174080, 1173790720, 6146048000, 32181125120, 168502558720, 882290851840, 4619734876160, 24189245849600, 126656535592960, 663182230159360

Offset: 1

R. H. Hardin, Mar 18 2013

nonn

Column 3 of A223186.

			Some solutions for n=3:
..0..7..1....0..2..6....0..5..7....0..2..6....0..5.10....0..6..4....0..7..1
..5..0..2....6..0..2....7.11..5....6..4..2....7.11..5....5.10..6....1..0..2
..7..1..0....2..1..0....5..7.11....2..6..4....5.10..6....6..5..0....2..1..8
Vertex neighbors:
0 -> 1 2 5 6 7
1 -> 0 2 3 7 8
2 -> 0 1 4 6 8
3 -> 1 7 8 9 11
4 -> 2 6 8 9 10
5 -> 0 6 7 10 11
6 -> 0 2 4 5 10
7 -> 0 1 3 5 11
8 -> 1 2 3 4 9
9 -> 3 4 8 10 11
10 -> 4 5 6 9 11
11 -> 3 5 7 9 10

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

Cf. A223186.

Empirical: a(n) = 6*a(n-1) - 4*a(n-2) for n>3.
Conjectures from Colin Barker, Aug 17 2018: (Start)
G.f.: 5*x*(5 - 14*x + 4*x^2) / (1 - 6*x + 4*x^2).
a(n) = sqrt(5)*((3-sqrt(5))^n*(1+sqrt(5)) + (-1+sqrt(5))*(3+sqrt(5))^n) for n>1.
(End)

A223182 Rolling icosahedron footprints: number of n X 4 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

Original entry on oeis.org

125, 320, 2080, 14560, 103520, 738720, 5274720, 37664800, 268947680, 1920431520, 13712917600, 97917648160, 699184991200, 4992559175840, 35649574015840, 254557248560160, 1817676496339680, 12979193733707680, 92678466337073760

Offset: 1

R. H. Hardin, Mar 18 2013

nonn

Column 4 of A223186.

			Some solutions for n=3:
..0..5..7.11....0..5.10.11....0..5..7..0....0..6..2..0....0..5.10.11
..7.11..3..7....7.11..5.10....7.11..5..7....2..0..1..7....7.11..9.10
..5..7..1..3....5..7.11..5....3..7..0..5....1..7..0..1....5.10..4..9
Vertex neighbors:
0 -> 1 2 5 6 7
1 -> 0 2 3 7 8
2 -> 0 1 4 6 8
3 -> 1 7 8 9 11
4 -> 2 6 8 9 10
5 -> 0 6 7 10 11
6 -> 0 2 4 5 10
7 -> 0 1 3 5 11
8 -> 1 2 3 4 9
9 -> 3 4 8 10 11
10 -> 4 5 6 9 11
11 -> 3 5 7 9 10

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

Cf. A223186.

Empirical: a(n) = 10*a(n-1) - 25*a(n-2) + 36*a(n-3) - 24*a(n-4) + 4*a(n-5) for n>6.

Empirical g.f.: 5*x*(25 - 186*x + 401*x^2 - 548*x^3 + 280*x^4 - 36*x^5) / ((1 - x)*(1 - 9*x + 16*x^2 - 20*x^3 + 4*x^4)). - Colin Barker, Aug 17 2018

A223183 Rolling icosahedron footprints: number of nX5 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

Original entry on oeis.org

625, 1280, 10880, 103520, 1018720, 10117600, 100734400, 1003652480, 10001217120, 99661921440, 993129226400, 9896517562400, 98618666649920, 982733746858880, 9792929198608800, 97586414236542560, 972447371181052000

Offset: 1

R. H. Hardin Mar 18 2013

nonn

Column 5 of A223186

			Some solutions for n=3
..0..6..2..0..1....0..1..2..6..0....0..5..6..0..2....0..7.11..3..7
..2..0..6..2..8....2..8..4..2..6....6..0..2..1..8....1..3..9.11..3
..1..2..0..1..3....1..2..6..0..2....2..1..8..3..1....7.11..3..9.11
Vertex neighbors:
0 -> 1 2 5 6 7
1 -> 0 2 3 7 8
2 -> 0 1 4 6 8
3 -> 1 7 8 9 11
4 -> 2 6 8 9 10
5 -> 0 6 7 10 11
6 -> 0 2 4 5 10
7 -> 0 1 3 5 11
8 -> 1 2 3 4 9
9 -> 3 4 8 10 11
10 -> 4 5 6 9 11
11 -> 3 5 7 9 10

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

Empirical: a(n) = 18*a(n-1) -114*a(n-2) +436*a(n-3) -1171*a(n-4) +2198*a(n-5) -2626*a(n-6) +1520*a(n-7) +79*a(n-8) -490*a(n-9) +156*a(n-10) +20*a(n-11) -16*a(n-12) for n>14

A223184 Rolling icosahedron footprints: number of nX6 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

Original entry on oeis.org

3125, 5120, 56960, 738720, 10117600, 141047120, 1978496760, 27809548920, 391129835720, 5502120113200, 77403634963000, 1088923059178480, 15319126239416880, 215511834396404680, 3031854469386423040

Offset: 1

R. H. Hardin Mar 18 2013

nonn

Column 6 of A223186

			Some solutions for n=3
..0..5..7..1..2..8....0..7..5..0..7..5....0..1..2..0..1..7....0..2..1..3..7.11
..7.11..3..8..1..2....1..0..7..1..0..7....2..0..1..2..0..1....1..0..7..1..3..7
..3..7..1..3..8..4....7..1..0..7..5..0....6..2..8..1..7..3....7..1..0..7.11..5
Vertex neighbors:
0 -> 1 2 5 6 7
1 -> 0 2 3 7 8
2 -> 0 1 4 6 8
3 -> 1 7 8 9 11
4 -> 2 6 8 9 10
5 -> 0 6 7 10 11
6 -> 0 2 4 5 10
7 -> 0 1 3 5 11
8 -> 1 2 3 4 9
9 -> 3 4 8 10 11
10 -> 4 5 6 9 11
11 -> 3 5 7 9 10

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

Empirical: a(n) = 34*a(n-1) -485*a(n-2) +4346*a(n-3) -28802*a(n-4) +150848*a(n-5) -636337*a(n-6) +2159092*a(n-7) -5838756*a(n-8) +12438350*a(n-9) -20709448*a(n-10) +27153588*a(n-11) -29621182*a(n-12) +31551558*a(n-13) -39983024*a(n-14) +57518400*a(n-15) -77018236*a(n-16) +86977128*a(n-17) -80388139*a(n-18) +58849554*a(n-19) -31645539*a(n-20) +11534636*a(n-21) -3954177*a(n-22) +4130000*a(n-23) -5090133*a(n-24) +4343740*a(n-25) -2741059*a(n-26) +1290058*a(n-27) -452154*a(n-28) +127180*a(n-29) -31889*a(n-30) +7256*a(n-31) -1428*a(n-32) +212*a(n-33) -16*a(n-34) for n>37

A223185 Rolling icosahedron footprints: number of nX7 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

Original entry on oeis.org

15625, 20480, 298240, 5274720, 100734400, 1978496760, 39288329040, 783477286880, 15649402486000, 312770780957680, 6252336965690240, 124994057100215760, 2498886647844053120, 49958192537794251840

Offset: 1

R. H. Hardin Mar 18 2013

nonn

Column 7 of A223186

			Some solutions for n=3
..0..1..8..3.11.10..4....0..1..7..3..1..2..8....0..1..2..8..4..2..8
..7..3..1..7..5..6.10....7..3..1..7..0..1..2....7..0..1..2..8..4..2
..1..7..3.11.10..5..6....1..8..3..1..2..0..1....1..2..8..4..2..6..0
Vertex neighbors:
0 -> 1 2 5 6 7
1 -> 0 2 3 7 8
2 -> 0 1 4 6 8
3 -> 1 7 8 9 11
4 -> 2 6 8 9 10
5 -> 0 6 7 10 11
6 -> 0 2 4 5 10
7 -> 0 1 3 5 11
8 -> 1 2 3 4 9
9 -> 3 4 8 10 11
10 -> 4 5 6 9 11
11 -> 3 5 7 9 10

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

Empirical: a(n) = 66*a(n-1) -1994*a(n-2) +38728*a(n-3) -560345*a(n-4) +6524130*a(n-5) -63671988*a(n-6) +531974880*a(n-7) -3848127650*a(n-8) +24271100034*a(n-9) -134248443158*a(n-10) +655104948054*a(n-11) -2840396229946*a(n-12) +11036021680176*a(n-13) -38791547925412*a(n-14) +124494074580576*a(n-15) -367351231567847*a(n-16) +999423446499110*a(n-17) -2499682520503839*a(n-18) +5690193562332578*a(n-19) -11551701732786876*a(n-20) +20071834370464908*a(n-21) -26876930663729930*a(n-22) +16317455183578594*a(n-23) +48424143854136547*a(n-24) -232211807643068642*a(n-25) +619464131345898580*a(n-26) -1282603906919244192*a(n-27) +2246605267792077403*a(n-28) -3489756978989727022*a(n-29) +4961464486889135750*a(n-30) -6469026702301613346*a(n-31) +7329127932627038108*a(n-32) -6107234316090609958*a(n-33) +1162350802936222707*a(n-34) +7797511221055222740*a(n-35) -18320112533950980909*a(n-36) +25782686710553708814*a(n-37) -26288270304669587613*a(n-38) +19774652510597920460*a(n-39) -10455502060923941526*a(n-40) +3846976781027716952*a(n-41) -2806689875314218244*a(n-42) +5990574561835742408*a(n-43) -9733337958117669182*a(n-44) +11123090360366078896*a(n-45) -9526133271383869478*a(n-46) +6085416185309130020*a(n-47) -2519532035859176458*a(n-48) +198152251562375604*a(n-49) +409467046281215433*a(n-50) +173910200548990740*a(n-51) -953018710281797197*a(n-52) +1224804507173323572*a(n-53) -932642136688428695*a(n-54) +434183935379320384*a(n-55) -72193818175133294*a(n-56) -49906923480155472*a(n-57) +20474695186042204*a(n-58) +35115723175174348*a(n-59) -54464059521030890*a(n-60) +40858228760018700*a(n-61) -19405207254479583*a(n-62) +5489913900342334*a(n-63) -535412982931541*a(n-64) +144399016026068*a(n-65) -656384732198270*a(n-66) +758260979482646*a(n-67) -514336442439891*a(n-68) +247075410873890*a(n-69) -93435139860166*a(n-70) +33799715845024*a(n-71) -15185583427369*a(n-72) +7872340486682*a(n-73) -3646561240884*a(n-74) +1378702706660*a(n-75) -440156846148*a(n-76) +126163892100*a(n-77) -32801670964*a(n-78) +7254467820*a(n-79) -1320516804*a(n-80) +209594608*a(n-81) -28206256*a(n-82) +1579488*a(n-83) +418368*a(n-84) -62144*a(n-85) +6848*a(n-86) -4224*a(n-87) +512*a(n-88) for n>93

A223180 Rolling icosahedron footprints: number of n X n 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

Original entry on oeis.org

1, 20, 400, 14560, 1018720, 141047120, 39288329040, 22231433842560, 25710310945249680, 61010814457361940400

Offset: 1

R. H. Hardin, Mar 18 2013

nonn

Diagonal of A223186.

			Some solutions for n=3
..0..7..1....0..5..7....0..1..7....0..1..7....0..2..6....0..7..1....0..7.11
..5..0..7....7..0..1....7..3.11....7..3..1....6..4..2....5..0..7....1..3..7
..7..1..3....1..7..3...11..7..5....1..8..3....2..6..4....7..5.11....7..1..0
Vertex neighbors:
0 -> 1 2 5 6 7
1 -> 0 2 3 7 8
2 -> 0 1 4 6 8
3 -> 1 7 8 9 11
4 -> 2 6 8 9 10
5 -> 0 6 7 10 11
6 -> 0 2 4 5 10
7 -> 0 1 3 5 11
8 -> 1 2 3 4 9
9 -> 3 4 8 10 11
10 -> 4 5 6 9 11
11 -> 3 5 7 9 10

Cf. A223186.

cp's OEIS Frontend

A223181 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, vertical or antidiagonal neighbor moves along an icosahedral edge.

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A223182 Rolling icosahedron footprints: number of n X 4 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

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A223183 Rolling icosahedron footprints: number of nX5 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

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A223184 Rolling icosahedron footprints: number of nX6 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

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A223185 Rolling icosahedron footprints: number of nX7 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

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Formula

A223180 Rolling icosahedron footprints: number of n X n 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, vertical or antidiagonal neighbor moves along an icosahedral edge.

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Author

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Examples

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