A207997 T(n,k) = number of n X k 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

1, 1, 1, 2, 3, 2, 4, 9, 9, 4, 8, 27, 41, 27, 8, 16, 81, 187, 187, 81, 16, 32, 243, 853, 1302, 853, 243, 32, 64, 729, 3891, 9075, 9075, 3891, 729, 64, 128, 2187, 17749, 63267, 96831, 63267, 17749, 2187, 128, 256, 6561, 80963, 441090, 1034073, 1034073, 441090, 80963

Offset: 1

Author

R. H. Hardin, Feb 22 2012

Keywords

• nonn
• tabl

Comments

Number of colorings of the grid graph P_n X P_k using a maximum of 3 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

Examples

			Table starts
..1....1.....2.......4.........8.........16...........32............64
..1....3.....9......27........81........243..........729..........2187
..2....9....41.....187.......853.......3891........17749.........80963
..4...27...187....1302......9075......63267.......441090.......3075255
..8...81...853....9075.....96831....1034073.....11045757.....117997043
.16..243..3891...63267...1034073...16932816....277458045....4547477370
.32..729.17749..441090..11045757..277458045...6978332618..175605187731
.64.2187.80963.3075255.117997043.4547477370.175605187731.6787438272198
...
Some solutions for n=4, k=3:
..0..1..2....0..1..0....0..1..0....0..1..2....0..1..2....0..1..2....0..1..0
..2..0..1....2..0..2....1..0..2....1..2..1....2..0..1....1..2..1....1..2..1
..0..2..0....0..1..0....2..1..0....0..1..2....0..2..0....0..1..2....2..0..2
..1..0..1....1..2..1....1..0..1....1..2..0....2..0..2....2..0..1....1..2..0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..544
• R. J. Mathar, Counting 2-way monotonic terrace forms over rectangular landscapes, vixra:1511.0225, eq. (33)-(35).
• Eric Weisstein's World of Mathematics, Grid Graph
• Eric Weisstein's World of Mathematics, Vertex Coloring
• Wikipedia, Graph Coloring

Crossrefs

Cf. A020698 (column 3), A078100 (column 4), A207994 (column 5), A207995 (column 6), A207996 (column 7).

Main diagonal is A207993.

Cf. A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).

Formula

2*T(n,m) = A078099(n,m) for m>1. - R. J. Mathar, Nov 23 2015