A208021 -id:A208021 - 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.

A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 24, 5, 0, 0, 0, 72, 120, 6, 0, 0, 0, 168, 6720, 360, 7, 0, 0, 0, 360, 935040, 126360, 840, 8, 0, 0, 0, 744, 325061760, 265035240, 1128960, 1680, 9, 0, 0, 0, 1512, 283192323840, 3322711053720, 17160407040, 6510000, 3024, 10

Offset: 1

Alois P. Heinz, May 04 2012

The square diagonal grid graph DG_(n,n) has n^2 = A000290(n) vertices and 2*(n-1)*(2*n-1) = A002943(n-1) edges; see A212208 for example. The chromatic polynomial of DG_(n,n) has n^2+1 = A002522(n) coefficients.

This graph is also called the king graph. - Andrew Howroyd, Jun 25 2017

			Square array A(n,k) begins:
  1,   0,       0,           0,                0, ...
  2,   0,       0,           0,                0, ...
  3,   0,       0,           0,                0, ...
  4,  24,      72,         168,              360, ...
  5, 120,    6720,      935040,        325061760, ...
  6, 360,  126360,   265035240,    3322711053720, ...
  7, 840, 1128960, 17160407040, 2949948395735040, ...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Eric Weisstein's World of Mathematics, King Graph
Wikipedia, Chromatic Polynomial

Columns 1-5 give: A000027, A052762 = 24*A000332, 24*A068250, 24*A068251, 24*A068252.
Rows n=1-16 give: A000007, A000038, 3*A000007, 4*A068293, 5*A068294, 6*A068295, 7*A068296, 8*A068297, 9*A068298, 10*A068299, 11*A068300, 12*A068301, 13*A068302, 14*A068303, 15*A068304, 16*A068305.
Cf. A000290, A002943, A212208, A208021.

A289136 Number of colorings of the n X n king graph up to permutation of the colors.

Original entry on oeis.org

1, 1, 270, 30066912, 3662498214110836, 978788002444637886853083440, 1017523795194980592656592864724960780556190, 6723457445689415320074916040888682277646129463792942126176174

Offset: 1

Andrew Howroyd, Jun 25 2017

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..11
Eric Weisstein's World of Mathematics, King Graph

Main diagonal of A208021.

Cf. A212208, A212209.

A208021 = Import["https://oeis.org/A208021/b208021.txt", "Table"][[All, 2]];
a[n_] := A208021[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Sep 23 2019 *)

A208018 Number of n X 3 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).

Original entry on oeis.org

2, 7, 270, 27093, 5252041, 1688298227, 819147302097, 562766251199622, 522687453434959180, 633631710314384052082, 975235769907492737264795, 1863659976446678796423595540, 4341202639797991970260869847084, 12137362722177750192276998364963783

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

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

Column 3 of A208021.

Terms a(14) and beyond from Andrew Howroyd, Jun 25 2017

A208019 Number of n X 4 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).

Original entry on oeis.org

5, 87, 27093, 30066912, 80318704605, 421673189900658, 3840411377316733859, 55738840149772892317910, 1212109474086951274342594323, 37684550247867169399313491413883, 1614317526939802578784333734266333582, 92481709137707355217408169152041892948306

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

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

Column 4 of A208021.

Terms a(8) and beyond from Andrew Howroyd, Jun 25 2017

A208020 Number of n X 5 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).

Original entry on oeis.org

15, 1657, 5252041, 80318704605, 3662498214110836, 387950830575476495043, 81811898713607232898679186, 30915192640251267259536636853106, 19388812191991157504565482403825817325, 19036318336696557481934278608884237849314405

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

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

Column 5 of A208021.

Terms a(6) and beyond from Andrew Howroyd, Jun 25 2017

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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A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).

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A289136 Number of colorings of the n X n king graph up to permutation of the colors.

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A208018 Number of n X 3 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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A208019 Number of n X 4 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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A208020 Number of n X 5 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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