%I #13 Jun 28 2017 02:10:44

%S 1,5,5,25,105,25,125,2205,2205,125,625,46305,194485,46305,625,3125,

%T 972405,17153945,17153945,972405,3125,15625,20420505,1513010465,

%U 6354787485,1513010465,20420505,15625,78125,428830605,133450391205

%N T(n,k) = number of n X k 0..5 arrays with no entry increasing mod 6 by 5 rightwards or downwards, starting with upper left zero.

%C 1/6 the number of 6-colorings of the grid graph P_n X P_k. - _Andrew Howroyd_, Jun 26 2017

%H Andrew Howroyd, <a href="/A222281/b222281.txt">Table of n, a(n) for n = 1..325</a> (terms 1..111 from R. H. Hardin)

%F T(n, k) = 5 * (24*A198982(n,k) - 12*A198715(n,k) - 8*A207997(n,k) - 3) for n*k > 1. - _Andrew Howroyd_, Jun 27 2017

%e Table starts

%e ........1................5......................25..........................125

%e ........5..............105....................2205........................46305

%e .......25.............2205..................194485.....................17153945

%e ......125............46305................17153945...................6354787485

%e ......625...........972405..............1513010465................2354171487645

%e .....3125.........20420505............133450391205..............872117822449905

%e ....15625........428830605..........11770577485085...........323081602357856985

%e ....78125.......9005442705........1038187247574145........119687637492011211885

%e ...390625.....189114296805.......91570083319317865......44339047670574481807485

%e ..1953125....3971400232905.....8076654937439905005...16425682631297501047982145

%e ..9765625...83399404891005...712376276332499775685.6084998755694142903356375385

%e .48828125.1751387502711105.62832938018547611186345

%e ...

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

%e ..0..0..0..0....0..0..0..0....0..0..0..0....0..3..0..0....0..0..0..0

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

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

%Y Columns 1-7 are A000351(n-1), 5*A009965(n-1), A222276, A222277, A222278, A222279, A222280.

%Y Main diagonal is A068256.

%Y Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A198982 (unlabeled 6 colorings), A222340 (7 colorings), A222462 (8 colorings).

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Feb 14 2013