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T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).
6

%I #9 Mar 15 2023 16:27:02

%S 1,2,2,5,7,5,15,67,67,15,52,1080,4192,1080,52,203,25287,587208,587208,

%T 25287,203,877,794545,147512732,953124784,147512732,794545,877,4140,

%U 31858034,58668937932,3410310590912,3410310590912,58668937932,31858034

%N T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

%H Andrew Howroyd, <a href="/A207981/b207981.txt">Table of n, a(n) for n = 1..496</a> (terms 1..40 from R. H. Hardin).

%e Table starts

%e ....1........2...........5............15............52...............203

%e ....2........7..........67..........1080.........25287............794545

%e ....5.......67........4192........587208.....147512732.......58668937932

%e ...15.....1080......587208.....953124784.3410310590912.22727131144960768

%e ...52....25287...147512732.3410310590912

%e ..203...794545.58668937932

%e ..877.31858034

%e .4140

%e ...

%e Some solutions for n=4 k=3

%e ..0..0..0....0..0..1....0..0..0....0..0..0....0..1..0....0..0..0....0..0..0

%e ..1..1..1....1..2..1....1..1..1....1..1..1....2..1..0....1..2..1....1..1..1

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

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

%Y Columns 1..4 are A000110, A207978, A207979, A207980.

%Y Main diagonal is A361452.

%Y Cf. A207868.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Feb 22 2012