A198900 -id:A198900 - cp's OEIS frontend

A198906 T(n,k) = number of n X k 0..4 arrays with values 0..4 introduced in row major order and no element equal to any horizontal or vertical neighbor.

1, 1, 1, 2, 4, 2, 5, 33, 33, 5, 15, 380, 1211, 380, 15, 51, 4801, 50384, 50384, 4801, 51, 187, 62004, 2125425, 6907736, 2125425, 62004, 187, 715, 804833, 89793204, 948656912, 948656912, 89793204, 804833, 715, 2795, 10459180, 3794115705

Offset: 1

R. H. Hardin, Oct 31 2011

Number of colorings of the grid graph P_n X P_k using a maximum of 5 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

			Table starts
.....1..........1...............2....................5
.....1..........4..............33..................380
.....2.........33............1211................50384
.....5........380...........50384..............6907736
....15.......4801.........2125425............948656912
....51......62004........89793204.........130292546801
...187.....804833......3794115705.......17895005957823
...715...10459180....160319061892.....2457786852894234
..2795..135958401...6774239755817...337564362706067534
.11051.1767426404.286243775060868.46362726246946052884
...
Some solutions with values 0 to 4 for n=6, k=4:
..0..1..0..1....0..1..0..1....0..1..0..1....0..1..0..1....0..1..0..1
..1..0..1..0....1..0..1..0....1..0..1..0....1..0..1..0....1..0..1..0
..0..1..0..2....0..1..0..2....0..1..0..2....0..1..0..2....0..1..0..2
..2..0..2..0....2..0..3..0....2..0..2..3....2..0..1..0....2..0..1..3
..3..2..1..4....0..1..0..4....0..4..0..2....3..2..4..3....0..3..4..2
..2..4..2..1....2..4..3..1....1..3..1..4....1..0..1..2....4..0..1..4

Andrew Howroyd, Table of n, a(n) for n = 1..378 (terms 1..127 from R. H. Hardin)

Columns 1-7 are A007581(n-2), A198900, A198901, A198902, A198903, A198904, A198905.
Main diagonal is A198899.
Cf. A207997 (3 colorings), A198715 (4 colorings), A222144 (labeled 5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).

A214141 T(n,k)=Number of 0..4 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..4 introduced in row major order.

Original entry on oeis.org

1, 1, 4, 4, 17, 33, 11, 257, 514, 380, 40, 3074, 28278, 16388, 4801, 147, 40434, 1101051, 3221873, 524296, 62004, 568, 522515, 47730973, 396246659, 367793014, 16777232, 804833, 2227, 6800539, 2000093424, 56449101747, 142612676441, 41989913081

Offset: 1

R. H. Hardin Jul 05 2012

Table starts
....1......1.........4...........11.............40...............147
....4.....17.......257.........3074..........40434............522515
...33....514.....28278......1101051.......47730973........2000093424
..380..16388...3221873....396246659....56449101747.....7658621867351
.4801.524296.367793014.142612676441.66761857485037.29325981412599886

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

R. H. Hardin, Table of n, a(n) for n = 1..111

Column 1 is A198900

Empirical for column k:
k=1: a(n) = 17*a(n-1) -55*a(n-2) +39*a(n-3)
k=2: a(n) = 34*a(n-1) -64*a(n-2)
k=3: a(n) = 129*a(n-1) -1759*a(n-2) +7575*a(n-3) -9064*a(n-4) +3120*a(n-5)
k=4: a(n) = 373*a(n-1) -4754*a(n-2) +15312*a(n-3)
k=5: (order 10)
k=6: (order 9)
Empirical for row n:
n=1: a(k)=6*a(k-1)-7*a(k-2)-6*a(k-3)+8*a(k-4)
n=2: a(k)=10*a(k-1)+50*a(k-2)-116*a(k-3)-361*a(k-4)+106*a(k-5)+312*a(k-6)
n=3: (order 15)
n=4: (order 37)

A214166 T(n,k)=Number of 0..5 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..5 introduced in row major order.

Original entry on oeis.org

1, 1, 4, 4, 18, 34, 11, 337, 902, 481, 41, 5994, 88261, 60320, 8731, 161, 121778, 7386816, 27240856, 4242606, 174454, 694, 2518082, 655418810, 9601970064, 8548472292, 300785428, 3603244, 3151, 52655411, 57661437162, 3598372134742

Offset: 1

R. H. Hardin Jul 05 2012

Table starts
....1.......1..........4.............11................41..................161
....4......18........337...........5994............121778..............2518082
...34.....902......88261........7386816.........655418810..........57661437162
..481...60320...27240856.....9601970064.....3598372134742.....1329144373535118
.8731.4242606.8548472292.12515731371696.19767477649307133.30641183868207736684

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

R. H. Hardin, Table of n, a(n) for n = 1..83

Column 1 is A198900

Empirical for column k:
k=1: a(n) = 32*a(n-1) -262*a(n-2) +672*a(n-3) -441*a(n-4)
k=2: a(n) = 84*a(n-1) -945*a(n-2) +1562*a(n-3)
k=3: a(n) = 370*a(n-1) -18411*a(n-2) +297448*a(n-3) -1839799*a(n-4) +4424682*a(n-5) -4113757*a(n-6) +1249468*a(n-7)
k=4: a(n) = 1402*a(n-1) -130492*a(n-2) +2979072*a(n-3) -15573492*a(n-4) +12571416*a(n-5)
k=5: (order 15)
Empirical for row n:
n=1: a(k)=10*a(k-1)-30*a(k-2)+20*a(k-3)+31*a(k-4)-30*a(k-5)
n=2: a(k)=21*a(k-1)+49*a(k-2)-959*a(k-3)-1869*a(k-4)+7679*a(k-5)+15051*a(k-6)-6741*a(k-7)-13230*a(k-8)
n=3: (order 22)
n=4: (order 60)

A206389 T(n,k)=Number of nXk 0..6 arrays with no element equal to another within a city block distance of two, and new values 0..6 introduced in row major order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 4, 2, 5, 33, 55, 33, 5, 15, 380, 1368, 1368, 380, 15, 52, 4801, 34442, 67716, 34442, 4801, 52, 202, 62004, 868994, 3328979, 3328979, 868994, 62004, 202, 855, 804833, 21916090, 164270604, 319902496, 164270604, 21916090, 804833, 855

Offset: 1

R. H. Hardin Feb 07 2012

Table starts
...1......1.........1..........2...........5..........15............52
...1......1.........4.........33.........380........4801.........62004
...1......4........55.......1368.......34442......868994......21916090
...2.....33......1368......67716.....3328979...164270604....8094257014
...5....380.....34442....3328979...319902496.30809397399.2965468376203
..15...4801....868994..164270604.30809397399
..52..62004..21916090.8094257014
.202.804833.552801398

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

R. H. Hardin, Table of n, a(n) for n = 1..60

Column 1 is A056272(n-2)

Column 2 is A198900(n-1)

cp's OEIS Frontend

A198906 T(n,k) = number of n X k 0..4 arrays with values 0..4 introduced in row major order and no element equal to any horizontal or vertical neighbor.

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A214141 T(n,k)=Number of 0..4 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..4 introduced in row major order.

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A214166 T(n,k)=Number of 0..5 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..5 introduced in row major order.

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Formula

A206389 T(n,k)=Number of nXk 0..6 arrays with no element equal to another within a city block distance of two, and new values 0..6 introduced in row major order.

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Examples

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