cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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).

Original entry on oeis.org

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

Views

Author

R. H. Hardin, Feb 21 2012

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Views

Author

R. H. Hardin, Feb 22 2012

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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).

Formula

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

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

Views

Author

R. H. Hardin, Oct 31 2011

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Views

Author

R. H. Hardin, Nov 01 2011

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Views

Author

R. H. Hardin, Oct 29 2011

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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).

A198914 T(n,k) = number of n X k 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, 1, 1, 2, 4, 2, 5, 34, 34, 5, 15, 500, 2051, 500, 15, 52, 10867, 269940, 269940, 10867, 52, 203, 313132, 54381563, 319608038, 54381563, 313132, 203, 877, 10856948, 13088156547, 481871809749, 481871809749, 13088156547, 10856948, 877, 4139
Offset: 1

Views

Author

R. H. Hardin, Oct 31 2011

Keywords

Comments

Number of colorings of the grid graph P_n X P_k using a maximum of 8 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

Examples

			Table starts
.....1............1..................2......................5
.....1............4.................34....................500
.....2...........34...............2051.................269940
.....5..........500.............269940..............319608038
....15........10867...........54381563...........481871809749
....52.......313132........13088156547........769126451071174
...203.....10856948......3352514013159....1243368053336112649
...877....418689772....876632051686733.2015791720035206825303
..4139..17067989413.230783525290600476
.21110.715189507700
...
Some solutions with values 0 to 7 for n=5, k=3:
..0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0
..1..2..1....1..0..1....1..2..1....1..0..2....1..0..2....1..2..3....1..2..3
..3..0..2....2..3..4....3..4..2....3..4..5....3..4..5....0..1..4....0..4..5
..2..4..5....5..4..3....5..6..1....5..3..6....6..7..0....5..6..7....1..5..1
..1..6..7....6..0..7....6..7..2....7..4..2....3..0..3....7..0..5....6..7..4

Links

Crossrefs

Columns 1-7 are A099262(n-1), A198908, A198909, A198910, A198911, A198912, A198913.
Main diagonal is A198907.
Cf. A207997 (3 colorings), A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A222462 (labeled 8 colorings), A207868 (unlimited).

A222444 T(n,k) = number of n X k 0..3 arrays with entries increasing mod 4 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

Original entry on oeis.org

1, 3, 3, 9, 21, 9, 27, 147, 147, 27, 81, 1029, 2403, 1029, 81, 243, 7203, 39285, 39285, 7203, 243, 729, 50421, 642249, 1500183, 642249, 50421, 729, 2187, 352947, 10499787, 57289767, 57289767, 10499787, 352947, 2187, 6561, 2470629, 171655443
Offset: 1

Views

Author

R. H. Hardin, Feb 20 2013

Keywords

Comments

1/4 the number of 4-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

Examples

			Table starts
......1..........3...............9..................27.......................81
......3.........21.............147................1029.....................7203
......9........147............2403...............39285...................642249
.....27.......1029...........39285.............1500183.................57289767
.....81.......7203..........642249............57289767...............5110723191
....243......50421........10499787..........2187822609.............455924913093
....729.....352947.......171655443.........83550197745...........40672916404629
...2187....2470629......2806303725.......3190677470643.........3628419487925547
...6561...17294403.....45878770089.....121847980727187.......323690312271131451
..19683..121060821....750047661027....4653221950068669.....28876324830999722133
..59049..847425747..12262131106083..177700725073710285...2576049100980154511889
.177147.5931980229.200467073061765.6786168386579878383.229808641254065144560647
...
Some solutions for n=3, k=4:
..0..0..0..2....0..0..2..0....0..2..0..0....0..2..0..2....0..0..2..3
..1..2..2..3....0..2..3..1....2..2..2..0....0..0..0..2....0..2..3..1
..2..2..3..1....2..0..1..3....2..2..0..0....2..0..1..3....1..2..0..1

Links

Crossrefs

Columns 1-7 are A000244(n-1), A169634(n-1), A222439, A222440, A222441, A222442, A222443.
Main diagonal is A068254.
Cf. A078099 (3 colorings), A198715 (unlabeled 4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A222462 (8 colorings).

Formula

T(n,k) = 6*A198715(n,k) - 3 for n*k>1. - Andrew Howroyd, Jun 27 2017
Empirical for column k:
k=1: a(n) = 3*a(n-1).
k=2: a(n) = 7*a(n-1).
k=3: a(n) = 18*a(n-1) - 27*a(n-2).
k=4: a(n) = 45*a(n-1) - 267*a(n-2) + 263*a(n-3).
k=5: a(n) = 118*a(n-1) - 2811*a(n-2) + 22255*a(n-3) - 53860*a(n-4) - 54747*a(n-5) + 269406*a(n-6) - 175392*a(n-7).
k=6: [order 13]
k=7: [order 32]

A222281 T(n,k) = number of n X k 0..5 arrays with no entry increasing mod 6 by 5 rightwards or downwards, starting with upper left zero.

Original entry on oeis.org

1, 5, 5, 25, 105, 25, 125, 2205, 2205, 125, 625, 46305, 194485, 46305, 625, 3125, 972405, 17153945, 17153945, 972405, 3125, 15625, 20420505, 1513010465, 6354787485, 1513010465, 20420505, 15625, 78125, 428830605, 133450391205
Offset: 1

Views

Author

R. H. Hardin, Feb 14 2013

Keywords

Comments

1/6 the number of 6-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

Examples

			Table starts
........1................5......................25..........................125
........5..............105....................2205........................46305
.......25.............2205..................194485.....................17153945
......125............46305................17153945...................6354787485
......625...........972405..............1513010465................2354171487645
.....3125.........20420505............133450391205..............872117822449905
....15625........428830605..........11770577485085...........323081602357856985
....78125.......9005442705........1038187247574145........119687637492011211885
...390625.....189114296805.......91570083319317865......44339047670574481807485
..1953125....3971400232905.....8076654937439905005...16425682631297501047982145
..9765625...83399404891005...712376276332499775685.6084998755694142903356375385
.48828125.1751387502711105.62832938018547611186345
...
Some solutions for n=3, k=4:
..0..0..0..0....0..0..0..0....0..0..0..0....0..3..0..0....0..0..0..0
..4..2..0..1....1..2..0..4....0..0..0..1....0..0..3..1....0..2..3..0
..0..4..1..4....1..4..1..2....3..4..4..1....3..0..4..4....4..5..1..3

Links

Crossrefs

Columns 1-7 are A000351(n-1), 5*A009965(n-1), A222276, A222277, A222278, A222279, A222280.
Main diagonal is A068256.
Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A198982 (unlabeled 6 colorings), A222340 (7 colorings), A222462 (8 colorings).

Formula

T(n, k) = 5 * (24*A198982(n,k) - 12*A198715(n,k) - 8*A207997(n,k) - 3) for n*k > 1. - Andrew Howroyd, Jun 27 2017

A222340 T(n,k) = number of n X k 0..6 arrays with no entry increasing mod 7 by 6 rightwards or downwards, starting with upper left zero.

Original entry on oeis.org

1, 6, 6, 36, 186, 36, 216, 5766, 5766, 216, 1296, 178746, 923526, 178746, 1296, 7776, 5541126, 147918906, 147918906, 5541126, 7776, 46656, 171774906, 23691810366, 122408393436, 23691810366, 171774906, 46656, 279936, 5325022086
Offset: 1

Views

Author

R. H. Hardin, Feb 15 2013

Keywords

Comments

1/7 the number of 7-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

Examples

			Table starts
.......1.............6...................36........................216
.......6...........186.................5766.....................178746
......36..........5766...............923526..................147918906
.....216........178746............147918906...............122408393436
....1296.......5541126..........23691810366............101297497221786
....7776.....171774906........3794659477146..........83827445649884946
...46656....5325022086......607781352505806.......69370328359709445996
..279936..165075684666....97346856728146986....57406526220963704077986
.1679616.5117346224646.15591808593304758846.47506035082750189614687546
...
Some solutions for n=3, k=4:
..0..0..2..0....0..2..2..0....0..0..0..0....0..2..0..0....0..2..0..0
..0..5..3..0....0..2..5..0....0..1..5..0....0..5..0..0....0..0..5..0
..3..1..4..5....4..4..2..3....3..6..1..4....2..2..2..2....4..1..3..4

Links

Crossrefs

Columns 1-6 are A000400(n-1), A222335, A222336, A222337, A222338, A222339.
Main diagonal is A068257.
Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A222281 (6 colorings), A198723 (unlabeled 7 colorings), A222462 (8 colorings).

Formula

T(n, k) = 6 * (120*A198723(n,k) - 60*A198906(n,k) - 40*A198715(n,k) - 15*A207997(n,k) - 4) for n*k > 1. - Andrew Howroyd, Jun 27 2017

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

Views

Author

R. H. Hardin, Feb 21 2013

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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).

Formula

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).
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