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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 horizontal, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).
10

%I #21 Mar 13 2023 13:31:21

%S 1,1,2,2,4,5,5,30,34,15,15,377,1573,500,52,52,7239,169064,192328,

%T 10900,203,203,193228,34438557,235862938,43366459,322768,877,877,

%U 6752442,11535531783,677352910013,746823293346,15769674462,12297768,4140

%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 horizontal, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

%H Andrew Howroyd, <a href="/A208096/b208096.txt">Table of n, a(n) for n = 1..210</a> (terms 1..39 from R. H. Hardin).

%e Table starts

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

%e ....2........4..........30..........377.........7239......193228.......6752442

%e ....5.......34........1573.......169064.....34438557.11535531783.5804338800147

%e ...15......500......192328....235862938.677352910013

%e ...52....10900....43366459.746823293346

%e ..203...322768.15769674462

%e ..877.12297768

%e .4140

%e Some solutions for n=4 k=3

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

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

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

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

%Y Columns 1..5 are A000110, A207864, A208093, A208094, A208095.

%Y Rows 1..5 are A000110(n-1), A208097, A208098, A208099, A208100.

%Y Main diagonal is A361449.

%Y Cf. A207868.

%K nonn,tabl

%O 1,3

%A _R. H. Hardin_, Feb 23 2012