A208054 -id:A208054 - cp's OEIS frontend

A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 48, 5, 0, 0, 6, 1056, 180, 6, 0, 0, 6, 45696, 32940, 480, 7, 0, 0, 6, 4034304, 30847500, 393600, 1050, 8, 0, 0, 6, 739642368, 148039757460, 3312560640, 2735250, 2016, 9

Offset: 1

Alois P. Heinz, May 02 2012

The rhombic hexagonal square grid graph RH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges; see A212162 for example. The chromatic polynomial of RH_(n,n) has n^2+1 = A002522(n) coefficients.

A differs from A212195 first at (n,k) = (4,5): A(4,5) = 4034304, A212195(4,5) = 4038432.

			Square array A(n,k) begins:
  1,    0,       0,            0,                 0, ...
  2,    0,       0,            0,                 0, ...
  3,    6,       6,            6,                 6, ...
  4,   48,    1056,        45696,           4034304, ...
  5,  180,   32940,     30847500,      148039757460, ...
  6,  480,  393600,   3312560640,   286169360240640, ...
  7, 1050, 2735250, 123791435250, 97337270132408250, ...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Wikipedia, Chromatic Polynomial

Columns k=1-6 give: A000027, A047927(n) = 6*A002417(n-2), 6*A068244, 6*A068245, 6*A068246, 6*A068247.
Rows n=1-15 give: A000007, A000038, A040006, 4*A068271, 5*A068272, 6*A068273, 7*A068274, 8*A068275, 9*A068276, 10*A068277, 11*A068278, 12*A068279, 13*A068280, 14*A068281, 15*A068282.
Cf. A212162, A212195, A208054, A208050.

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

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.

A208050 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 8, 8, 5, 14, 32, 44, 32, 14, 41, 128, 244, 244, 128, 41, 122, 512, 1356, 1904, 1356, 512, 122, 365, 2048, 7540, 14976, 14976, 7540, 2048, 365, 1094, 8192, 41932, 118096, 168096, 118096, 41932, 8192, 1094, 3281, 32768, 233204, 931968

Offset: 1

R. H. Hardin, Feb 22 2012

Equivalently, the number of colorings in the rhombic hexagonal square grid graph RH_(n,k) using 4 colors up to permutation of the colors. - Andrew Howroyd, Jun 25 2017

			Table starts
...1....1......2.......5........14.........41..........122...........365
...1....2......8......32.......128........512.........2048..........8192
...2....8.....44.....244......1356.......7540........41932........233204
...5...32....244....1904.....14976.....118096.......931968.......7356288
..14..128...1356...14976....168096....1897888.....21472544.....243113056
..41..512...7540..118096...1897888...30818432....502504448....8206614784
.122.2048..41932..931968..21472544..502504448..11838995200..279733684992
.365.8192.233204.7356288.243113056.8206614784.279733684992.9578237457408
...
Some solutions for n=4 k=3
..0..1..0....0..1..2....0..1..0....0..1..0....0..1..2....0..1..2....0..1..2
..2..3..1....2..3..0....2..3..1....2..3..1....2..0..3....2..3..0....2..0..3
..1..2..0....0..1..2....0..2..3....0..2..3....1..2..1....1..2..1....1..2..0
..3..1..2....2..3..1....1..0..1....3..0..2....3..0..3....3..0..2....3..1..2

R. H. Hardin, Table of n, a(n) for n = 1..312

Columns 1-7 are A007051(n-2), A004171(n-2), A208044, A208046, A208047-A208049.
Main diagonal is A208045.
Cf. A208054, A068271, A212162, A212163.

A208051 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 or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

2, 15, 716, 83440, 18171918, 6423127757, 3376465219485, 2486327138729353, 2454996141671330713, 3143641561804718358884, 5084413934903476204054959, 10166624612435356414640715276, 24691194409056203926713355839051

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..0....0..1..0....0..1..0....0..1..0
..2..3..1....2..3..1....2..3..2....2..3..1....2..3..2....2..3..4....2..3..1
..0..2..0....0..4..5....0..4..0....1..0..2....0..1..4....0..5..0....4..5..3
..1..3..1....5..3..4....1..2..1....2..3..0....2..0..1....1..2..1....0..6..4

Andrew Howroyd, Table of n, a(n) for n = 1..100 (first 20 terms from R. H. Hardin)

Column 3 of A208054.

Column 6 of A216460 and column 6 of A216612.

A208052 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 or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

5, 203, 83440, 112073062, 346212384169, 2043836452962923, 20538493062072286871, 324446721789278481458200, 7600607444778834885508561198, 252518539060608935260165703671351, 11485037650037932819944784728914383376, 694873103685620467779932914106484450991930

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

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

Column 4 of A208054.

Terms a(11) and beyond from Andrew Howroyd, Jun 25 2017

A208053 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 or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

15, 4140, 18171918, 346212384169, 18633407199331522, 2254561102047453593406, 531301410253869099621138608, 220853044418109083337335323829443, 150574889720989154216201888327789028154, 159239694548038760947916228538237933340567164

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

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

Column 5 of A208054.

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

A361451 Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any horizontal, vertical or antidiagonal neighbor.

Original entry on oeis.org

1, 2, 716, 112073062, 18633407199331522, 6575857942770612176290018153, 8769438200005128572266011359369913757287151, 72530091349507692706447958441062812294511923156598114466468667, 5746371835090565784276352813398004749296101606959968049467898643632416711373273639694

Offset: 1

Andrew Howroyd, Mar 13 2023

nonn

Main diagonal of A208054.

cp's OEIS Frontend

A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

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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).

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A208050 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208051 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 or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208052 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 or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208053 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 or antidiagonal neighbor (colorings ignoring permutations of colors).

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A361451 Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any horizontal, vertical or antidiagonal neighbor.

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