A289136 -id:A289136 - cp's OEIS frontend

A208021 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, vertical, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

1, 1, 1, 2, 1, 2, 5, 7, 7, 5, 15, 87, 270, 87, 15, 52, 1657, 27093, 27093, 1657, 52, 203, 43833, 5252041, 30066912, 5252041, 43833, 203, 877, 1515903, 1688298227, 80318704605, 80318704605, 1688298227, 1515903, 877, 4140, 65766991

Offset: 1

R. H. Hardin, Feb 22 2012

Equivalently, the number of colorings of the n x k king graph using any number of colors up to permutation of the colors. - Andrew Howroyd, Jun 25 2017

			Table starts
...1........1............2...............5...............15..............52
...1........1............7..............87.............1657...........43833
...2........7..........270...........27093..........5252041......1688298227
...5.......87........27093........30066912......80318704605.421673189900658
..15.....1657......5252041.....80318704605.3662498214110836
..52....43833...1688298227.421673189900658
.203..1515903.819147302097
.877.65766991
...
Some solutions for n=4 k=3
..0..1..2....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..2
..2..3..4....2..3..2....2..3..2....2..3..2....2..3..2....2..3..2....2..3..0
..5..6..0....4..0..4....0..1..0....4..1..0....0..1..0....0..4..0....4..5..4
..2..3..1....1..2..1....2..3..4....5..2..3....2..4..2....1..2..1....0..1..2

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

Columns 1-5 are A000110(n-1), A020556(n-1), A208018, A208019, A208020.
Main diagonal is A289136.
Cf. A207868, A212208, A212209.

A361453 Number of colorings of the n X n knight graph up to permutation of the colors.

Original entry on oeis.org

1, 15, 4141, 450288795, 50602429743064097, 12123635532529660182357354372

Offset: 1

Andrew Howroyd, Mar 13 2023

Any number of colors may be used.

Equivalently, a(n) is the number of stable partitions of the n X n knight graph. A stable partition is a partition of the vertices into sets so that no two vertices in a set are adjacent in the graph.

			a(2) = 15 = A000110(4) because the graph has no edges and so there are no restrictions on how the vertices may be colored (or equivalently the vertices partitioned into sets).

Eric Weisstein's World of Mathematics, Knight Graph.
Eric Weisstein's World of Mathematics, Vertex Coloring.

Main diagonal of A208001.

Cf. A000110, A207863 (grid graph), A289136 (king), A295178.

a(n) <= A000110(n^2).

A295177 Chromatic invariant of the n X n king graph.

Original entry on oeis.org

1, 2, 48, 31328, 473555616, 155337138737984, 1070159970425515438368, 152060635426686819239358391520, 440726752780548858707046038903585268960, 25867445082047583284454362951686346866758358099808

Offset: 1

Eric W. Weisstein, Nov 16 2017

Eric Weisstein's World of Mathematics, Chromatic Invariant
Eric Weisstein's World of Mathematics, King Graph

Cf. A182517, A212208, A212209, A289136.

a(6)-a(10) from Andrew Howroyd, Apr 23 2018

cp's OEIS Frontend

A208021 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, vertical, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

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A361453 Number of colorings of the n X n knight graph up to permutation of the colors.

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A295177 Chromatic invariant of the n X n king graph.

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