A223331 -id:A223331 - cp's OEIS frontend

A223327 Rolling cube footprints: number of n X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

1, 27, 3969, 3176523, 13876429707, 330815891296611, 43041050285035823937, 30560274879452174084575347, 118414345277803156878479453881947, 2503923842303442402830521022775619070739

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Diagonal of A223331

			Some solutions for n=3
..0..1..3....0..1..5....0..4..5....0..1..0....0..1..5....0..1..0....0..2..3
..0..2..6....5..1..5....0..4..5....5..4..6....5..4..0....0..1..5....3..1..3
..6..4..5....3..7..3....6..4..5....0..2..6....6..4..6....0..1..5....0..1..3
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

R. H. Hardin, Table of n, a(n) for n = 1..13

A223328 Rolling cube footprints: number of n X 5 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

Original entry on oeis.org

81, 9261, 1059723, 121264857, 13876429707, 1587890407761, 181703507374179, 20792470582897209, 2379298227030964827, 272264906211251105313, 31155480347969275662483, 3565145318286297427548489

Offset: 1

R. H. Hardin, Mar 19 2013

nonn

Column 5 of A223331.

			Some solutions for n=3:
..0..2..0..2..6....0..4..6..4..0....0..4..5..7..6....0..4..0..2..3
..0..2..0..2..6....6..2..0..4..0....0..4..6..4..6....6..2..6..2..0
..0..2..6..4..0....0..2..0..1..3....0..2..0..2..0....0..2..3..2..0
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

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

Cf. A223331.

Empirical: a(n) = 117*a(n-1) - 294*a(n-2).

Empirical g.f.: 27*x*(3 - 8*x) / (1 - 117*x + 294*x^2). - Colin Barker, Aug 19 2018

A223329 Rolling cube footprints: number of n X 6 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

Original entry on oeis.org

243, 64827, 17324685, 4630596579, 1237689159021, 330815891296611, 88422165323034669, 23633929115222765283, 6316997592145707860589, 1688439462802532886238947, 451294745322310558858483245

Offset: 1

R. H. Hardin, Mar 19 2013

nonn

Column 6 of A223331.

			Some solutions for n=3:
..0..4..0..4..6..2....0..4..0..1..0..1....0..4..6..7..5..1....0..4..0..1..5..4
..0..4..0..2..0..2....0..4..5..4..5..4....0..4..5..4..0..1....0..4..5..4..6..7
..0..4..6..4..0..2....0..4..5..4..5..4....0..4..5..4..0..2....0..4..5..4..5..7
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

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

Cf. A223331.

Empirical: a(n) = 282*a(n-1) - 3969*a(n-2) + 9604*a(n-3).

Empirical g.f.: 27*x*(9 - 137*x + 294*x^2) / (1 - 282*x + 3969*x^2 - 9604*x^3). - Colin Barker, Aug 19 2018

A223330 Rolling cube footprints: number of nX7 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

Original entry on oeis.org

729, 453789, 283231809, 176834343105, 110409049892673, 68935627430614161, 43041050285035823937, 26873361387913413198609, 16778808787792177773698625, 10476115000476214621652175825, 6540928315682905702406399281857

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Column 7 of A223331

			Some solutions for n=3
..0..4..0..4..5..4..5....0..4..0..4..6..7..3....0..4..0..4..0..4..0
..0..4..0..1..5..1..3....0..4..0..2..3..1..5....0..4..0..4..6..2..0
..0..4..5..4..5..1..5....0..4..6..7..5..7..6....0..4..6..2..6..2..0
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

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

Empirical: a(n) = 692*a(n-1) -43569*a(n-2) +847042*a(n-3) -6303164*a(n-4) +15731352*a(n-5)

A223332 Rolling cube footprints: number of 3 X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

Original entry on oeis.org

64, 243, 3969, 64827, 1059723, 17324685, 283231809, 4630406067, 75700050363, 1237579942725, 20232537609249, 330771018512907, 5407599817782603, 88405979220238365, 1445302430884160289, 23628482316968449347

Offset: 1

R. H. Hardin, Mar 19 2013

nonn

Row 3 of A223331.

			Some solutions for n=3:
  0 2 6    0 4 6    0 2 6    0 1 5    0 2 3    0 1 5    0 2 6
  6 7 6    6 2 3    3 7 5    5 4 6    3 7 6    5 7 3    0 4 0
  3 7 6    6 2 0    6 4 6    6 2 0    6 7 3    6 7 6    6 4 6
Vertex neighbors:
  0 -> 1 2 4
  1 -> 0 3 5
  2 -> 0 3 6
  3 -> 1 2 7
  4 -> 0 5 6
  5 -> 1 4 7
  6 -> 2 4 7
  7 -> 3 5 6

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

Cf. A223331.

Empirical: a(n) = 18*a(n-1) - 27*a(n-2) for n>4.
Conjectures from Colin Barker, Aug 19 2018: (Start)
G.f.: x*(64 - 909*x + 1323*x^2 - 54*x^3) / (1 - 18*x + 27*x^2).
a(n) = (49/54)*((9-3*sqrt(6))^n + (3*(3+sqrt(6)))^n) for n>2.
(End)

A223333 Rolling cube footprints: number of 4 X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

Original entry on oeis.org

512, 2187, 83349, 3176523, 121264857, 4630596579, 176834343105, 6753068175483, 257891143282857, 9848539671395859, 376103406869296785, 14362918587487614123, 548501892736263190137, 20946601106278812695619

Offset: 1

R. H. Hardin, Mar 19 2013

nonn

			Some solutions for n=3:
..0..2..0....0..1..0....0..2..3....0..1..0....0..1..3....0..4..5....0..1..0
..6..2..3....0..1..0....6..7..3....0..2..0....3..1..0....5..4..5....5..4..5
..6..7..5....0..1..3....3..7..6....3..1..3....5..1..0....0..1..0....0..1..0
..3..1..3....5..1..0....6..2..6....0..1..0....3..1..5....3..2..6....0..4..5
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

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

Row 4 of A223331.

Empirical: a(n) = 48*a(n-1) - 402*a(n-2) + 1064*a(n-3) - 789*a(n-4) for n>7.

Empirical g.f.: x*(512 - 22389*x + 184197*x^2 - 489823*x^3 + 375051*x^4 - 112104*x^5 + 121716*x^6) / ((1 - 3*x)*(1 - 45*x + 267*x^2 - 263*x^3)). - Colin Barker, Aug 19 2018

A223334 Rolling cube footprints: number of 5Xn 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

Original entry on oeis.org

4096, 19683, 1750329, 155649627, 13876429707, 1237689159021, 110409049892673, 9849460912651899, 878665830242295627, 78385506226393458141, 6992749153173468569913, 623821244550672567090699

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Row 5 of A223331

			Some solutions for n=3
..0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0
..0..4..6....0..2..6....6..4..6....0..4..5....0..4..6....0..2..3....0..4..0
..0..4..0....3..2..0....6..7..5....5..4..6....5..4..5....3..7..3....6..2..3
..0..4..5....0..2..3....6..4..0....6..7..5....5..1..0....3..1..0....6..2..0
..6..4..5....6..7..3....0..2..6....5..1..5....3..2..6....3..1..3....6..2..0
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

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

Empirical: a(n) = 131*a(n-1) -4376*a(n-2) +62456*a(n-3) -430316*a(n-4) +1335338*a(n-5) -688543*a(n-6) -5374827*a(n-7) +10631682*a(n-8) -5437152*a(n-9) for n>13

A223335 Rolling cube footprints: number of 6 X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

Original entry on oeis.org

32768, 177147, 36756909, 7626831723, 1587890407761, 330815891296611, 68935627430614161, 14365712340521444763, 2993767914167348634225, 623894511848537009674251, 130018376961215856234304281

Offset: 1

R. H. Hardin, Mar 19 2013

nonn

Row 6 of A223331.

			Some solutions for n=3
..0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0
..0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0
..0..2..6....0..2..6....0..4..0....0..4..6....0..2..6....0..4..6....0..2..0
..6..4..0....3..7..5....0..2..0....6..4..0....6..7..5....6..2..6....6..2..0
..0..1..0....3..1..0....0..1..3....6..2..0....5..4..0....3..2..6....6..2..6
..5..4..5....0..2..0....0..2..0....6..4..6....0..2..3....0..2..6....0..4..0
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

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

Cf. A223331.

Empirical: a(n) = 363*a(n-1) -42090*a(n-2) +2394790*a(n-3) -76949379*a(n-4) +1439989707*a(n-5) -14188685367*a(n-6) +22105928025*a(n-7) +1145452404696*a(n-8) -13381549738272*a(n-9) +56254263011793*a(n-10) +62916731731323*a(n-11) -1574353040753800*a(n-12) +5560131978318054*a(n-13) -1833699655619436*a(n-14) -34189681765260294*a(n-15) +75258675804889728*a(n-16) -6463571598539280*a(n-17) -143095150109785392*a(n-18) +141900806436429312*a(n-19) -28670292192657024*a(n-20) for n>25.

A223336 Rolling cube footprints: number of 7Xn 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

Original entry on oeis.org

262144, 1594323, 771895089, 373714754427, 181703507374179, 88422165323034669, 43041050285035823937, 20952806226803446547979, 10200311867482640569384803, 4965790918632215208497280621, 2417489227433689564468440912057

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Row 7 of A223331

			Some solutions for n=3
..0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0
..0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0
..0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0....0..4..0
..0..4..6....0..4..5....0..2..6....0..4..0....0..2..0....0..2..3....6..2..6
..5..4..5....6..4..5....0..2..6....0..4..5....3..1..5....6..2..6....6..7..3
..6..4..6....5..1..5....6..2..6....5..4..6....3..1..3....6..2..0....3..2..3
..6..4..0....3..7..5....6..2..0....6..2..0....3..2..3....3..2..6....3..7..3
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

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

Empirical: a(n) = 1017*a(n-1) -381153*a(n-2) +76330925*a(n-3) -9379669404*a(n-4) +747532248534*a(n-5) -38499863181596*a(n-6) +1139954614587576*a(n-7) -5583055176706479*a(n-8) -1094431324506682925*a(n-9) +50537184076461725934*a(n-10) -1015258431965719361244*a(n-11) +3095570263329310857414*a(n-12) +351032728024216265247750*a(n-13) -8715911720129630011302099*a(n-14) +67896300297745233456034095*a(n-15) +862567216416574756707869697*a(n-16) -25376301969118851499248569685*a(n-17) +189481741633620976743863445870*a(n-18) +1213881680830549398947848361250*a(n-19) -33701546470498136557101837454170*a(n-20) +193817191950113071661498477713494*a(n-21) +1105391498469154893964681382128230*a(n-22) -21334493200414307906998488358434330*a(n-23) +79226547906556204139213650387509229*a(n-24) +554852186342095607470419069220373331*a(n-25) -6242703657785811406542257712875475819*a(n-26) +11427072259363660331384127096122658759*a(n-27) +130251581424149170693623570641906239872*a(n-28) -791454998632744644019698078348974703252*a(n-29) +150183001780574465288696031737669683446*a(n-30) +13021503011401984451434439252140174731984*a(n-31) -39238347770704076075162550684665356370931*a(n-32) -54363423267461595132483159664169706700377*a(n-33) +511583255147743885153317284758744493422968*a(n-34) -611466515286479680979853300688193412132840*a(n-35) -2288068847976359716858330082703129157497324*a(n-36) +7143935107773190831122804291852527329425180*a(n-37) -646497305327209377732210692074196271867879*a(n-38) -24260350279666085911492453378468843083738153*a(n-39) +31366989158217866691843797464920854600869029*a(n-40) +18403204408585929605908614199621520086661637*a(n-41) -72616510295082242362026374686186931457144498*a(n-42) +46505758190559990385450468895760155088783636*a(n-43) +28528493644475533596840131207619521026564086*a(n-44) -60080596653327296067550212394399506018872694*a(n-45) +38580578456372532586316733205323399729445890*a(n-46) -12050175487651135339486502499983588770170012*a(n-47) +1509805849627288567831603949640808080305292*a(n-48) +117034612538436236819170072080398495133180*a(n-49) -52192142460391429881159477398798743239984*a(n-50) +4001693815102511240259579194781596297544*a(n-51) for n>57

cp's OEIS Frontend

A223327 Rolling cube footprints: number of n X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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A223328 Rolling cube footprints: number of n X 5 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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A223329 Rolling cube footprints: number of n X 6 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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A223330 Rolling cube footprints: number of nX7 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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A223332 Rolling cube footprints: number of 3 X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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A223333 Rolling cube footprints: number of 4 X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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A223334 Rolling cube footprints: number of 5Xn 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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A223335 Rolling cube footprints: number of 6 X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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A223336 Rolling cube footprints: number of 7Xn 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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