A198914 -id:A198914 - cp's OEIS frontend

A207868 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

1, 1, 1, 2, 4, 2, 5, 34, 34, 5, 15, 500, 2052, 500, 15, 52, 10900, 278982, 278982, 10900, 52, 203, 322768, 68162042, 455546040, 68162042, 322768, 203, 877, 12297768, 26419793726, 1625686993918, 1625686993918, 26419793726, 12297768, 877

Offset: 1

R. H. Hardin, Feb 21 2012

Table starts
...1.........1..............2.................5.................15
...1.........4.............34...............500..............10900
...2........34...........2052............278982...........68162042
...5.......500.........278982.........455546040......1625686993918
..15.....10900.......68162042.....1625686993918.103204230192540988
..52....322768....26419793726.10764437129618296
.203..12297768.15002771641712
.877.580849872

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

Andrew Howroyd, Table of n, a(n) for n = 1..231 (terms 1..49 from R. H. Hardin)
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Vertex Coloring.
Wikipedia, Graph Coloring.

Columns 1..5 are A000110(n-1), A207864, A207865, A207866, A207867.
Main diagonal is A207863.
Cf. A207997 (3 colorings), A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings).
Cf. A207981, A208001 (knight), A208021 (king), A208054, A208096, A208301.

A207997 T(n,k) = number of n X k 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 9, 9, 4, 8, 27, 41, 27, 8, 16, 81, 187, 187, 81, 16, 32, 243, 853, 1302, 853, 243, 32, 64, 729, 3891, 9075, 9075, 3891, 729, 64, 128, 2187, 17749, 63267, 96831, 63267, 17749, 2187, 128, 256, 6561, 80963, 441090, 1034073, 1034073, 441090, 80963

Offset: 1

R. H. Hardin, Feb 22 2012

Number of colorings of the grid graph P_n X P_k using a maximum of 3 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

			Table starts
..1....1.....2.......4.........8.........16...........32............64
..1....3.....9......27........81........243..........729..........2187
..2....9....41.....187.......853.......3891........17749.........80963
..4...27...187....1302......9075......63267.......441090.......3075255
..8...81...853....9075.....96831....1034073.....11045757.....117997043
.16..243..3891...63267...1034073...16932816....277458045....4547477370
.32..729.17749..441090..11045757..277458045...6978332618..175605187731
.64.2187.80963.3075255.117997043.4547477370.175605187731.6787438272198
...
Some solutions for n=4, k=3:
..0..1..2....0..1..0....0..1..0....0..1..2....0..1..2....0..1..2....0..1..0
..2..0..1....2..0..2....1..0..2....1..2..1....2..0..1....1..2..1....1..2..1
..0..2..0....0..1..0....2..1..0....0..1..2....0..2..0....0..1..2....2..0..2
..1..0..1....1..2..1....1..0..1....1..2..0....2..0..2....2..0..1....1..2..0

R. H. Hardin, Table of n, a(n) for n = 1..544
R. J. Mathar, Counting 2-way monotonic terrace forms over rectangular landscapes, vixra:1511.0225, eq. (33)-(35).
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Wikipedia, Graph Coloring

Cf. A020698 (column 3), A078100 (column 4), A207994 (column 5), A207995 (column 6), A207996 (column 7).
Main diagonal is A207993.
Cf. A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).

2*T(n,m) = A078099(n,m) for m>1. - R. J. Mathar, Nov 23 2015

A198715 T(n,k)=Number of nXk 0..3 arrays with values 0..3 introduced in row major order and no element equal to any horizontal or vertical neighbor.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 5, 25, 25, 5, 14, 172, 401, 172, 14, 41, 1201, 6548, 6548, 1201, 41, 122, 8404, 107042, 250031, 107042, 8404, 122, 365, 58825, 1749965, 9548295, 9548295, 1749965, 58825, 365, 1094, 411772, 28609241, 364637102, 851787199, 364637102

Offset: 1

R. H. Hardin, Oct 29 2011

Number of colorings of the grid graph P_n X P_k using a maximum of 4 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

			Table starts
....1........1............2...............5..................14
....1........4...........25.............172................1201
....2.......25..........401............6548..............107042
....5......172.........6548..........250031.............9548295
...14.....1201.......107042.........9548295...........851787199
...41.....8404......1749965.......364637102.........75987485516
..122....58825.....28609241.....13925032958.......6778819400772
..365...411772....467717288....531779578441.....604736581320925
.1094..2882401...7646461682..20307996787865...53948385378521909
.3281.20176804.125007943505.775536991678112.4812720805166620356
...
Some solutions with all values from 0 to 3 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..2..1....0..1..0..1....0..1..0..1....0..1..0..2....0..1..0..1
..1..2..0..3....2..0..3..0....2..0..1..0....1..2..1..3....1..2..3..0
..2..0..2..0....1..3..0..2....3..2..0..2....0..3..0..2....3..1..2..3
..3..2..0..1....3..2..1..0....0..3..2..1....3..1..3..0....1..3..1..0

Andrew Howroyd, Table of n, a(n) for n = 1..496 (terms 1..180 from R. H. Hardin)
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Wikipedia, Graph Coloring

Columns 1-7 are A007051(n-2), A034494(n-1), A198710, A198711, A198712, A198713, A198714.
Main diagonal is A198709.
Cf. A207997 (3 colorings), A222444 (labeled 4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).

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.

Original entry on oeis.org

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).

A198982 T(n,k) = number of n X k 0..5 arrays with values 0..5 introduced in row major order and no element equal to any horizontal or vertical neighbor.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 5, 34, 34, 5, 15, 481, 1835, 481, 15, 52, 8731, 146286, 146286, 8731, 52, 202, 174454, 12662226, 53082012, 12662226, 174454, 202, 855, 3603244, 1112962873, 19622872903, 19622872903, 1112962873, 3603244, 855, 3845, 75251971

Offset: 1

R. H. Hardin, Nov 01 2011

Number of colorings of the grid graph P_n X P_k using a maximum of 6 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

			Table starts
.....1...........1.................2........................5
.....1...........4................34......................481
.....2..........34..............1835...................146286
.....5.........481............146286.................53082012
....15........8731..........12662226..............19622872903
....52......174454........1112962873............7267830860056
...202.....3603244.......98102456246.........2692353648978984
...855....75251971.....8651794282083.......997397244990907738
..3845..1577395861...763087851014929....369492074075459555844
.18002.33105096904.67305520316532514.136880688981914387733120
...
Some solutions with all values from 0 to 5 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..2..3..0....1..2..3..0....1..2..3..0....1..2..3..0....1..2..3..0
..0..3..0..1....0..3..0..1....0..3..0..1....0..3..0..1....0..3..0..1
..2..0..3..0....1..0..3..0....1..0..3..0....1..0..3..0....1..0..3..0
..1..2..0..2....4..1..0..1....3..1..0..4....3..4..2..1....3..4..2..5
..4..5..2..1....5..4..2..4....5..2..3..1....1..5..1..5....5..1..5..2

Andrew Howroyd, Table of n, a(n) for n = 1..325 (terms 1..111 from R. H. Hardin)

Columns 1-7 are A056272(n-1), A198976, A198977, A198978, A198979, A198980, A198981.
Main diagonal is A198975.
Cf. A207997 (3 colorings), A198715 (4 colorings), A198906 (5 colorings), A222281 (labeled 6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).

A198723 T(n,k) = number of n X k 0..6 arrays with values 0..6 introduced in row major order and no element equal to any horizontal or vertical neighbor.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 5, 34, 34, 5, 15, 499, 2027, 499, 15, 52, 10507, 232841, 232841, 10507, 52, 203, 272410, 34003792, 173549032, 34003792, 272410, 203, 876, 7817980, 5315840795, 141168480719, 141168480719, 5315840795, 7817980, 876, 4111

Offset: 1

R. H. Hardin, Oct 29 2011

Number of colorings of the grid graph P_n X P_k using a maximum of 7 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

			Table starts
.....1............1...................2.......................5
.....1............4..................34.....................499
.....2...........34................2027..................232841
.....5..........499..............232841...............173549032
....15........10507............34003792............141168480719
....52.......272410..........5315840795.........116492275674072
...203......7817980........846047363854.......96356630422085931
...876....234638905.....135284283124811....79732515488691835557
..4111...7176366133...21658679381667910.65980773070548173552412
.20648.221220625936.3468618095206638077
...
Some solutions with all values 0 to 6 for n=3, k=3:
..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..0....0..1..2
..3..2..4....2..3..1....3..4..5....1..3..4....3..4..3....2..3..4....3..4..3
..4..5..6....4..5..6....6..2..4....5..0..6....1..5..6....5..4..6....5..6..2

Andrew Howroyd, Table of n, a(n) for n = 1..325 (terms 1..84 from R. H. Hardin)

Columns 1-7 are A056273(n-1), A198717, A198718, A198719, A198720, A198721, A198722.
Main diagonal is A198716.
Cf. A207997 (3 colorings), A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A222340 (labeled 7 colorings), A198914 (8 colorings), A207868 (unlimited).

A222462 T(n,k) = number of n X k 0..7 arrays with no entry increasing mod 8 by 7 rightwards or downwards, starting with upper left zero.

Original entry on oeis.org

1, 7, 7, 49, 301, 49, 343, 12943, 12943, 343, 2401, 556549, 3418807, 556549, 2401, 16807, 23931607, 903055069, 903055069, 23931607, 16807, 117649, 1029059101, 238535974201, 1465295106499, 238535974201, 1029059101, 117649, 823543

Offset: 1

R. H. Hardin, Feb 21 2013

1/8 the number of 8-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
......1.............7..................49........................343
......7...........301...............12943.....................556549
.....49.........12943.............3418807..................903055069
....343........556549...........903055069..............1465295106499
...2401......23931607........238535974201...........2377584520856755
..16807....1029059101......63007686842527........3857863258420747009
.117649...44249541343...16643060295393343.....6259760185235726701945
.823543.1902730277749.4396153388210813341.10157072698503130798653535
...
Some solutions for n=3, k=4:
..0..4..2..3....0..0..0..4....0..4..6..1....0..4..0..4....0..2..6..2
..0..0..5..6....0..0..4..6....0..0..1..5....0..0..6..0....0..0..2..3
..0..0..0..1....0..0..5..1....0..0..3..5....0..0..0..1....0..0..3..5

Andrew Howroyd, Table of n, a(n) for n = 1..276 (terms 1..83 from R. H. Hardin)

Columns 1-5 are A000420(n-1), 7*43^(n-1), A222459, A222460, A222461.
Main diagonal is A068258.
Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A198914 (unlabeled 8 colorings).

T(n, k) = 7 * (720*A198914(n,k) - 360*A198982(n,k) - 240*A198906(n,k) - 90*A198715(n,k) - 24*A207997(n,k) - 5) for n*k > 1. - Andrew Howroyd, Jun 27 2017
Empirical for column k:
k=1: a(n) = 7*a(n-1).
k=2: a(n) = 43*a(n-1).
k=3: a(n) = 270*a(n-1) - 1547*a(n-2).
k=4: a(n) = 1689*a(n-1) - 108775*a(n-2) + 1672631*a(n-3).
k=5: a(n) = 10754*a(n-1) - 8060499*a(n-2) + 2219242223*a(n-3) - 245682627864*a(n-4) + 5798947687589*a(n-5) + 448113231493438*a(n-6) - 2763020698450992*a(n-7).

A198908 Number of n X 2 0..7 arrays with values 0..7 introduced in row major order and no element equal to any horizontal or vertical neighbor.

Original entry on oeis.org

1, 4, 34, 500, 10867, 313132, 10856948, 418689772, 17067989413, 715189507700, 30371156968582, 1298083132473604, 55654030558406039, 2389712969490386908, 102686352402421016536, 4414019789796312628796, 189771542890022982723145

Offset: 1

R. H. Hardin, Oct 31 2011

nonn

Column 2 of A198914.

			Some solutions with values 0 to 7 for n=5:
..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..2..3....2..3....1..2....2..3....2..3....2..3....2..3....2..3....2..3....2..3
..4..5....4..2....3..4....4..5....4..5....4..5....4..5....4..5....4..5....4..5
..0..3....1..5....5..6....2..3....6..7....3..6....6..2....6..1....0..6....6..7
..6..7....6..7....2..7....6..7....7..1....7..3....2..7....5..7....7..1....2..4

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A198914.

Empirical: a(n) = 88*a(n-1) - 2613*a(n-2) + 33232*a(n-3) - 184531*a(n-4) + 400344*a(n-5) - 246519*a(n-6).
Conjectures from Colin Barker, Feb 22 2018: (Start)
G.f.: x*(1 - 84*x + 2295*x^2 - 25272*x^3 + 107312*x^4 - 128772*x^5) / ((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 13*x)*(1 - 21*x)*(1 - 43*x)).
a(n) = (1036945 + 344344*3^n + 75465*7^n + 12040*13^n + 2795*21^n + 91*43^n) / 2817360.
(End)

A198907 Number of n X n 0..7 arrays with values 0..7 introduced in row major order and no element equal to any horizontal or vertical neighbor.

Original entry on oeis.org

1, 4, 2051, 319608038, 4717456775959530, 2869765342757082981876352

Offset: 1

R. H. Hardin Oct 31 2011

nonn

Diagonal of A198914

			Some solutions with values 0 to 7 for n=3
..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
..3..4..5....3..4..5....3..4..5....3..4..1....3..4..5....3..4..5....3..4..5
..6..7..4....1..6..7....4..6..7....5..6..7....2..6..7....6..0..7....6..2..7

Jean-François Alcover, Table of n, a(n) for n = 1..12

A198914 = Cases[Import["https://oeis.org/A198914/b198914.txt", "Table"], {, }][[All, 2]];
a[n_] := A198914[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Sep 23 2019 *)

A198909 Number of nX3 0..7 arrays with values 0..7 introduced in row major order and no element equal to any horizontal or vertical neighbor.

Original entry on oeis.org

2, 34, 2051, 269940, 54381563, 13088156547, 3352514013159, 876632051686733, 230783525290600476, 60892298359864756053, 16078352442288906898825, 4246466444504000973551502, 1121629692628918068552449185

Offset: 1

R. H. Hardin Oct 31 2011

nonn

Column 3 of A198914

			Some solutions with values 0 to 7 for n=4
..0..1..2....0..1..2....0..1..2....0..1..0....0..1..2....0..1..2....0..1..2
..3..4..0....3..4..5....3..0..4....2..3..4....1..3..4....3..0..4....3..4..0
..5..3..6....4..2..0....5..4..6....3..5..6....5..0..6....4..1..5....5..6..3
..0..1..7....1..6..7....6..1..7....7..4..0....0..1..7....6..7..4....3..5..7

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n) = 431*a(n-1) -53038*a(n-2) +2570412*a(n-3) -55789402*a(n-4) +610919758*a(n-5) -3707175288*a(n-6) +13010264300*a(n-7) -26380804149*a(n-8) +29517406483*a(n-9) -16243626186*a(n-10) +3246286680*a(n-11)

cp's OEIS Frontend

A207868 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A207997 T(n,k) = number of n X k 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A198715 T(n,k)=Number of nXk 0..3 arrays with values 0..3 introduced in row major order and no element equal to any horizontal or vertical neighbor.

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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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A198982 T(n,k) = number of n X k 0..5 arrays with values 0..5 introduced in row major order and no element equal to any horizontal or vertical neighbor.

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A198723 T(n,k) = number of n X k 0..6 arrays with values 0..6 introduced in row major order and no element equal to any horizontal or vertical neighbor.

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A222462 T(n,k) = number of n X k 0..7 arrays with no entry increasing mod 8 by 7 rightwards or downwards, starting with upper left zero.

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A198908 Number of n X 2 0..7 arrays with values 0..7 introduced in row major order and no element equal to any horizontal or vertical neighbor.

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A198907 Number of n X n 0..7 arrays with values 0..7 introduced in row major order and no element equal to any horizontal or vertical neighbor.

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Mathematica

A198909 Number of nX3 0..7 arrays with values 0..7 introduced in row major order and no element equal to any horizontal or vertical neighbor.

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