%I #13 Jun 28 2017 02:11:00

%S 1,7,7,49,301,49,343,12943,12943,343,2401,556549,3418807,556549,2401,

%T 16807,23931607,903055069,903055069,23931607,16807,117649,1029059101,

%U 238535974201,1465295106499,238535974201,1029059101,117649,823543

%N 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.

%C 1/8 the number of 8-colorings of the grid graph P_n X P_k. - _Andrew Howroyd_, Jun 26 2017

%H Andrew Howroyd, <a href="/A222462/b222462.txt">Table of n, a(n) for n = 1..276</a> (terms 1..83 from R. H. Hardin)

%F 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

%F Empirical for column k:

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

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

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

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

%F 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).

%e Table starts

%e ......1.............7..................49........................343

%e ......7...........301...............12943.....................556549

%e .....49.........12943.............3418807..................903055069

%e ....343........556549...........903055069..............1465295106499

%e ...2401......23931607........238535974201...........2377584520856755

%e ..16807....1029059101......63007686842527........3857863258420747009

%e .117649...44249541343...16643060295393343.....6259760185235726701945

%e .823543.1902730277749.4396153388210813341.10157072698503130798653535

%e ...

%e Some solutions for n=3, k=4:

%e ..0..4..2..3....0..0..0..4....0..4..6..1....0..4..0..4....0..2..6..2

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

%e ..0..0..0..1....0..0..5..1....0..0..3..5....0..0..0..1....0..0..3..5

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

%Y Main diagonal is A068258.

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

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Feb 21 2013