A264945 -id:A264945 - cp's OEIS frontend

A264939 Number of n X n arrays containing n copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

1, 2, 56, 184740, 137409486772, 36724170744914466520, 5395269899168064415277849230476

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Diagonal of A264945.

			Some solutions for n=4
..0..1..2..0....0..1..2..0....0..1..0..1....0..1..2..0....0..1..0..2
..3..2..0..3....0..2..3..1....2..1..3..0....3..2..1..2....2..1..2..3
..3..2..1..3....1..2..1..3....2..3..2..3....0..3..0..1....0..1..0..3
..0..1..2..1....2..3..0..3....2..3..0..1....3..1..3..2....2..3..1..3

Cf. A264945.

A264946 Number of 3 X n arrays containing n copies of 0..3-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 8, 56, 332, 2350, 16108, 114148, 817280, 5918424, 43251920, 318428920, 2359455400, 17577965926, 131579085320, 989014916960, 7461197116280, 56471149527616, 428656384570808, 3262347081071272, 24887490475059512

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Row 3 of A264945.

			Some solutions for n=4:
  0 1 2 1    0 1 0 1    0 1 0 2    0 1 2 1    0 1 2 0
  1 0 2 0    0 2 1 2    1 0 2 1    2 0 2 1    0 2 1 2
  1 2 0 2    2 0 2 1    2 0 1 2    1 0 2 0    1 0 1 2

R. H. Hardin, Table of n, a(n) for n = 1..104
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Cf. A264945.

Conjectured recurrence of order 9 and degree 13: (n + 7)*(n + 9)*(n + 11)*(n + 12)*(1125*n^9 + 71100*n^8 + 1849290*n^7 + 25782456*n^6 + 208721101*n^5 + 972463852*n^4 + 2219593700*n^3 + 50298752*n^2 - 10311481536*n - 14857032960)*a(n + 9) - (n + 8)*(n + 11)*(1125*n^11 + 105975*n^10 + 4287165*n^9 + 97635351*n^8 + 1380358747*n^7 + 12566316081*n^6 + 73253605103*n^5 + 255547606333*n^4 + 398433897060*n^3 - 409020958588*n^2 - 2573789550288*n - 2893020641280)*a(n + 8) - (n + 7)*(55125*n^12 + 5043150*n^11 + 203593785*n^10 + 4783109874*n^9 + 72494563771*n^8 + 740695604282*n^7 + 5151899227595*n^6 + 23797245731102*n^5 + 66502991649164*n^4 + 73821686951912*n^3 - 148787634331808*n^2 - 594255830565888*n - 585045220193280)*a(n + 7) + (-111375*n^13 - 10629900*n^12 - 452242260*n^11 - 11349359364*n^10 - 187042020462*n^9 - 2127929201500*n^8 - 17043729555112*n^7 - 95651673757276*n^6 - 362118971182795*n^5 - 819923758737640*n^4 - 569404973885116*n^3 + 2298397321892016*n^2 + 6690359386550016*n + 5779315376271360)*a(n + 6) + 4*(68625*n^13 + 6350850*n^12 + 262268790*n^11 + 6397257006*n^10 + 102643250092*n^9 + 1138907266426*n^8 + 8907524302014*n^7 + 48769985579898*n^6 + 178866084976275*n^5 + 380439835948764*n^4 + 162302253918524*n^3 - 1409022793603840*n^2 - 3584273436805056*n - 2901502849512960)*a(n + 5) + 8*(113625*n^13 + 10109475*n^12 + 400010115*n^11 + 9340598391*n^10 + 143726469497*n^9 + 1536021852345*n^8 + 11649644036681*n^7 + 62442624371309*n^6 + 227373943553018*n^5 + 494618567601008*n^4 + 295071232164264*n^3 - 1523160592469296*n^2 - 4224581839405248*n - 3597712690682880)*a(n + 4) + 16*(10125*n^13 + 791775*n^12 + 27734085*n^11 + 599490219*n^10 + 9285055875*n^9 + 110260636645*n^8 + 1005826848007*n^7 + 6794028883657*n^6 + 32142152171668*n^5 + 97251455731576*n^4 + 145564079336272*n^3 - 68830984119216*n^2 - 616978454170176*n - 699609347458560)*a(n + 3) - 16*(102375*n^13 + 8429850*n^12 + 302519490*n^11 + 6252315786*n^10 + 82728789292*n^9 + 735119632054*n^8 + 4454619907050*n^7 + 18120079094814*n^6 + 45886145527965*n^5 + 51009713975544*n^4 - 78118093654492*n^3 - 396941915101552*n^2 - 632960801821824*n - 397766739363840)*a(n + 2) - 64*(n + 1)*(28125*n^11 + 2145375*n^10 + 69983925*n^9 + 1286257695*n^8 + 14737434403*n^7 + 109491713225*n^6 + 526473720215*n^5 + 1544097805013*n^4 + 2140227079172*n^3 - 1382380121116*n^2 - 9705594256208*n - 11040556116480)*n*a(n + 1) - 512*(n - 1)*(n + 1)*(1125*n^9 + 81225*n^8 + 2458590*n^7 + 40812786*n^6 + 406374277*n^5 + 2472650097*n^4 + 8781110488*n^3 + 15058677204*n^2 + 1549576672*n - 21689733120)*n^2*a(n) = 0. - Manuel Kauers and Christoph Koutschan, Mar 06 2023

Conjecture: a(n) ~ 3^(7/2) * 2^(3*n - 5) / (Pi*n), based on the recurrence by Manuel Kauers and Christoph Koutschan. - Vaclav Kotesovec, Mar 07 2023

A264947 Number of 4 X n arrays containing n copies of 0..4-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 60, 3201, 184740, 11375145, 730983420, 48402531561, 3282992503164, 226854309720993, 15915758107113276, 1130694005695927761, 81177583723495750340, 5880587303767912833417, 429300706847441007321756

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Row 4 of A264945.

			Some solutions for n=4:
  0 1 0 2   0 1 2 0   0 1 2 0   0 1 0 1   0 1 2 0
  2 0 3 1   3 0 1 2   0 3 1 3   0 2 3 1   3 1 0 3
  2 3 2 1   3 2 3 1   3 2 0 1   3 1 3 2   2 3 1 2
  1 3 0 3   1 2 3 0   2 1 2 3   2 0 2 3   3 1 0 2

Christoph Koutschan, Table of n, a(n) for n = 1..80 (first 20 terms from R. H. Hardin).
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Cf. A264945.

cols = Tuples[{0, 1, 2, 3}, 4];
tmat = Table[If[Or @@ MapThread[SameQ, cols[[{i, j}]]], 0, 1], {i, 256}, {j, 256}];
vec = vvec = ((x^Count[#, 0] * y^Count[#, 1] * z^Count[#, 2]) & /@ cols);
Prepend[Table[vec = Expand[vvec*(tmat.vec)]; Coefficient[Total[vec], (x*y*z)^n]/24, {n, 2, 10}], 1]

A264941 Number of nX3 arrays containing 3 copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

0, 1, 56, 3201, 307016, 43200625, 8403205056, 2162875665601, 712215325718576, 292162814336854881, 146127700393451369000

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Column 3 of A264945.

			Some solutions for n=4
..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
..3..0..2....3..0..2....3..2..0....3..0..1....0..1..3....3..1..3....1..2..3
..3..1..3....3..1..3....0..1..3....2..3..0....1..2..3....2..3..0....2..1..3
..1..0..2....0..2..1....1..3..2....1..3..2....2..3..0....2..1..0....0..3..0

Cf. A264945.

A264942 Number of nX4 arrays of permutations of 4 copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

0, 2, 332, 184740, 196389904, 368696849320, 1122409297218768, 5194172319787113488, 34756519036284839519360

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Column 4 of A264945.

			Some solutions for n=4
..0..1..2..0....0..1..2..0....0..1..2..0....0..1..0..2....0..1..0..2
..1..2..3..2....1..2..0..3....1..3..0..2....1..2..3..0....1..3..1..0
..0..1..3..2....3..1..2..3....3..1..3..2....1..2..3..0....3..0..2..1
..3..1..3..0....2..0..3..1....1..3..2..0....3..1..3..2....3..2..3..2

Cf. A264945.

A264943 Number of nX5 arrays of permutations of 5 copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

0, 1, 2350, 11375145, 137409486772, 3537141103745065, 173357342638694530002, 14844736259829478729001905

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Column 5 of A264945.

			Some solutions for n=3
..0..1..2..0..1....0..1..2..1..2....0..1..2..1..2....0..1..0..2..1
..0..2..0..1..2....1..2..1..0..2....2..0..2..0..1....1..2..0..1..0
..1..2..1..0..2....0..2..0..1..0....1..0..2..0..1....2..0..2..1..2

Cf. A264945.

A264944 Number of nX6 arrays of permutations of 6 copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

0, 2, 16108, 730983420, 102117302729744, 36724170744914466520, 29526013062983124375791088

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Column 6 of A264945.

			Some solutions for n=3
..0..1..0..2..0..2....0..1..2..0..2..1....0..1..0..1..0..2....0..1..2..1..0..2
..0..1..0..1..2..1....1..0..1..2..1..0....2..0..2..0..1..2....0..1..0..2..0..1
..2..1..2..1..2..0....2..0..2..1..2..0....1..2..0..1..2..1....2..0..2..1..2..1

Cf. A264945.

A264948 Number of 5Xn arrays containing n copies of 0..5-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 544, 307016, 196389904, 137409486772, 102117302729744, 79300380540248536, 63677294433618454384

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Row 5 of A264945.

			Some solutions for n=2
..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..2..3....2..3....2..3....2..3....2..3....0..2....2..3....2..3....2..3....2..3
..4..2....4..0....0..4....0..4....2..0....1..3....0..4....0..1....1..0....4..3
..1..3....1..2....2..3....2..1....4..3....2..4....1..4....2..4....3..4....0..4
..0..4....4..3....4..1....3..4....1..4....4..3....3..2....3..4....4..2....2..1

Cf. A264945.

A264949 Number of 6Xn arrays containing n copies of 0..6-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 6040, 43200625, 368696849320, 3537141103745065, 36724170744914466520

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Row 6 of A264945.

			Some solutions for n=2
..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..2..3....2..3....2..0....1..0....2..3....2..0....2..3....2..3....2..3....2..1
..3..4....4..5....3..4....2..3....4..1....2..3....4..2....2..0....0..4....3..2
..1..5....4..0....4..5....2..4....5..3....4..1....4..3....4..1....5..2....0..3
..5..0....5..1....5..1....4..5....0..4....5..3....5..0....5..3....3..5....4..5
..2..4....2..3....2..3....3..5....5..2....5..4....1..5....5..4....1..4....5..4

Cf. A264945.

A264950 Number of 7Xn arrays containing n copies of 0..7-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 79008, 8403205056, 1122409297218768, 173357342638694530002, 29526013062983124375791088, 5395269899168064415277849230476

Offset: 1

R. H. Hardin, Nov 29 2015

nonn

Row 7 of A264945.

			Some solutions for n=2
..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..2..0....2..3....1..2....2..3....0..1....2..3....2..3....2..3....2..3....2..3
..3..4....4..2....3..4....4..0....2..3....1..3....4..5....4..5....2..0....4..3
..5..6....5..0....5..0....5..4....4..5....4..0....3..4....4..5....4..5....5..1
..3..2....6..5....6..3....2..5....6..2....5..4....6..2....1..6....3..1....2..6
..6..1....6..4....5..2....6..1....5..6....6..5....5..0....2..0....6..4....5..4
..5..4....3..1....4..6....6..3....3..4....6..2....6..1....3..6....5..6....6..0

Cf. A264945.

cp's OEIS Frontend

A264939 Number of n X n arrays containing n copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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A264946 Number of 3 X n arrays containing n copies of 0..3-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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A264947 Number of 4 X n arrays containing n copies of 0..4-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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Mathematica

A264941 Number of nX3 arrays containing 3 copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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A264942 Number of nX4 arrays of permutations of 4 copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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A264943 Number of nX5 arrays of permutations of 5 copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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A264944 Number of nX6 arrays of permutations of 6 copies of 0..n-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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A264948 Number of 5Xn arrays containing n copies of 0..5-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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A264949 Number of 6Xn arrays containing n copies of 0..6-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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A264950 Number of 7Xn arrays containing n copies of 0..7-1 with no equal horizontal neighbors and new values introduced sequentially from 0.

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