A208001 -id:A208001 - 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.

A207998 Number of nX2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any knight-move neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

2, 15, 114, 1657, 36401, 1094076, 42436913, 2042803419, 118645071202, 8138047375093, 648222594284197, 59148486677131940, 6113429673418115581, 708972517372901061559, 91508202542991877872978, 13053785934982985242672065

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 2 of A208001.

Terms a(16) and beyond from Andrew Howroyd, Jul 01 2017

A207999 Number of nX3 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any knight-move neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

5, 114, 4141, 426422, 86545486, 29169661126, 14823734802701, 10648697150313375, 10321527909363771779, 13032394645031912097587, 20852865726433378283251197, 41354508341339617041441302471, 99804566252370862012176094395266, 288659258928461717629676421776606809

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..50

Column 3 of A208001.

Terms a(8) and beyond from Andrew Howroyd, Jul 01 2017

A208000 Number of nX4 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any knight-move neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

15, 1657, 426422, 450288795, 1182700979380, 6184248124218800, 56513375910372605548, 826725568737937330039810, 18171823971396715681105965282, 572057633247843506849301892746850, 24840621984533068280588275487942834483, 1443495070617319156132945205013820325621405

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..50

Column 4 of A208001.

Terms a(6) and beyond from Andrew Howroyd, Jul 01 2017

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).

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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A207998 Number of nX2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any knight-move neighbor (colorings ignoring permutations of colors).

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A207999 Number of nX3 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any knight-move neighbor (colorings ignoring permutations of colors).

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A208000 Number of nX4 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any knight-move 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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