A198715 T(n,k)=Number of nXk 0..3 arrays with values 0..3 introduced in row major order and no element equal to any horizontal or vertical neighbor.

1, 1, 1, 2, 4, 2, 5, 25, 25, 5, 14, 172, 401, 172, 14, 41, 1201, 6548, 6548, 1201, 41, 122, 8404, 107042, 250031, 107042, 8404, 122, 365, 58825, 1749965, 9548295, 9548295, 1749965, 58825, 365, 1094, 411772, 28609241, 364637102, 851787199, 364637102

Offset

1,4

Comments

Number of colorings of the grid graph P_n X P_k using a maximum of 4 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

Links

Andrew Howroyd, Table of n, a(n) for n = 1..496 (terms 1..180 from R. H. Hardin)

Eric Weisstein's World of Mathematics, Grid Graph

Eric Weisstein's World of Mathematics, Vertex Coloring

Wikipedia, Graph Coloring

Example

Table starts
....1........1............2...............5..................14
....1........4...........25.............172................1201
....2.......25..........401............6548..............107042
....5......172.........6548..........250031.............9548295
...14.....1201.......107042.........9548295...........851787199
...41.....8404......1749965.......364637102.........75987485516
..122....58825.....28609241.....13925032958.......6778819400772
..365...411772....467717288....531779578441.....604736581320925
.1094..2882401...7646461682..20307996787865...53948385378521909
.3281.20176804.125007943505.775536991678112.4812720805166620356
...
Some solutions with all values from 0 to 3 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..2..1....0..1..0..1....0..1..0..1....0..1..0..2....0..1..0..1
..1..2..0..3....2..0..3..0....2..0..1..0....1..2..1..3....1..2..3..0
..2..0..2..0....1..3..0..2....3..2..0..2....0..3..0..2....3..1..2..3
..3..2..0..1....3..2..1..0....0..3..2..1....3..1..3..0....1..3..1..0

Crossrefs

Columns 1-7 are A007051(n-2), A034494(n-1), A198710, A198711, A198712, A198713, A198714.

Main diagonal is A198709.

Cf. A207997 (3 colorings), A222444 (labeled 4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).

Sequence in context: A055883 A366588 A085843 * A216663 A198906 A198982

Adjacent sequences: A198712 A198713 A198714 * A198716 A198717 A198718

Keyword

nonn,tabl

Author

R. H. Hardin, Oct 29 2011

Status

approved