A222462 T(n,k) = number of n X k 0..7 arrays with no entry increasing mod 8 by 7 rightwards or downwards, starting with upper left zero.

1, 7, 7, 49, 301, 49, 343, 12943, 12943, 343, 2401, 556549, 3418807, 556549, 2401, 16807, 23931607, 903055069, 903055069, 23931607, 16807, 117649, 1029059101, 238535974201, 1465295106499, 238535974201, 1029059101, 117649, 823543

Offset: 1

Author

R. H. Hardin, Feb 21 2013

Keywords

• nonn
• tabl

Comments

1/8 the number of 8-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

Examples

			Table starts
......1.............7..................49........................343
......7...........301...............12943.....................556549
.....49.........12943.............3418807..................903055069
....343........556549...........903055069..............1465295106499
...2401......23931607........238535974201...........2377584520856755
..16807....1029059101......63007686842527........3857863258420747009
.117649...44249541343...16643060295393343.....6259760185235726701945
.823543.1902730277749.4396153388210813341.10157072698503130798653535
...
Some solutions for n=3, k=4:
..0..4..2..3....0..0..0..4....0..4..6..1....0..4..0..4....0..2..6..2
..0..0..5..6....0..0..4..6....0..0..1..5....0..0..6..0....0..0..2..3
..0..0..0..1....0..0..5..1....0..0..3..5....0..0..0..1....0..0..3..5
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..276 (terms 1..83 from R. H. Hardin)

Crossrefs

Columns 1-5 are A000420(n-1), 7*43^(n-1), A222459, A222460, A222461.

Main diagonal is A068258.

Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A198914 (unlabeled 8 colorings).

Formula

T(n, k) = 7 * (720*A198914(n,k) - 360*A198982(n,k) - 240*A198906(n,k) - 90*A198715(n,k) - 24*A207997(n,k) - 5) for n*k > 1. - Andrew Howroyd, Jun 27 2017

Empirical for column k:

k=1: a(n) = 7*a(n-1).

k=2: a(n) = 43*a(n-1).

k=3: a(n) = 270*a(n-1) - 1547*a(n-2).

k=4: a(n) = 1689*a(n-1) - 108775*a(n-2) + 1672631*a(n-3).

k=5: a(n) = 10754*a(n-1) - 8060499*a(n-2) + 2219242223*a(n-3) - 245682627864*a(n-4) + 5798947687589*a(n-5) + 448113231493438*a(n-6) - 2763020698450992*a(n-7).