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T(n,k)=Number of nXk 0..k arrays with every repeated value in every row and column equal to the previous repeated value, and new values introduced in row-major sequential order.
6

%I #4 Jan 24 2016 09:14:31

%S 1,2,2,5,14,4,14,187,122,7,45,3552,11051,938,12,163,93311,1880990,

%T 605969,6734,20,657,3201247,555258347,982227549,30785259,45938,33,

%U 2910,137687080,256785961187,3511764501993,477324637497,1470330667,302402,54

%N T(n,k)=Number of nXk 0..k arrays with every repeated value in every row and column equal to the previous repeated value, and new values introduced in row-major sequential order.

%C Table starts

%C ..1.......2...........5..............14................45...............163

%C ..2......14.........187............3552.............93311...........3201247

%C ..4.....122.......11051.........1880990.........555258347......256785961187

%C ..7.....938......605969.......982227549.....3511764501993.23932385578547037

%C .12....6734....30785259....477324637497.20888913167378133

%C .20...45938..1470330667.217487430054270

%C .33..302402.67008101445

%C .54.1939154

%C .88

%H R. H. Hardin, <a href="/A268003/b268003.txt">Table of n, a(n) for n = 1..45</a>

%F Empirical for column k:

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

%F k=2: a(n) = 14*a(n-1) -60*a(n-2) +50*a(n-3) +145*a(n-4) -80*a(n-5) -84*a(n-6) +16*a(n-7)

%F k=3: [order 20]

%e Some solutions for n=3 k=4

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

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

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

%Y Column 1 is A000071(n+2).

%Y Column 2 is A267906.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Jan 24 2016