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T(n,k)=Number of nXk 0..k arrays with every repeated value in every row equal to, and in every column unequal to, the previous repeated value, and new values introduced in row-major sequential order.
4

%I #4 Jan 28 2016 07:35:37

%S 1,2,2,5,14,3,14,187,96,5,45,3552,9054,726,7,163,93311,1589578,494098,

%T 5046,11,657,3201247,479973420,829256141,25770278,35574,15,2910,

%U 137687080,225814538887,3038628153922,423155007379,1339895662,242406,23

%N T(n,k)=Number of nXk 0..k arrays with every repeated value in every row equal to, and in every column unequal 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 ..3......96........9054.........1589578.........479973420......225814538887

%C ..5.....726......494098.......829256141.....3038628153922.21090319968167260

%C ..7....5046....25770278....423155007379.19146833412153174

%C .11...35574..1339895662.214576839826736

%C .15..242406.68390077014

%C .23.1653750

%C .31

%H R. H. Hardin, <a href="/A268180/b268180.txt">Table of n, a(n) for n = 1..45</a>

%F Empirical for column k:

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

%F k=2: [order 6] for n>8

%F k=3: [order 17] for n>19

%e Some solutions for n=3 k=4

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

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

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

%Y Column 1 is A052955(n-1).

%Y Column 2 is A267913.

%Y Row 1 is A268004.

%Y Row 2 is A268005.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Jan 28 2016