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T(n,k) = Number of (n+1) X (k+1) 0..2 arrays with every 2 X 2 subblock sum differing from its neighbors horizontally, vertically, diagonally and antidiagonally.
6

%I #6 Apr 22 2021 14:18:08

%S 81,558,558,3844,3600,3844,26784,23546,23546,26784,186624,167832,

%T 140536,167832,186624,1296000,1184928,1029888,1029888,1184928,1296000,

%U 9000000,8243488,7431136,8153960,7431136,8243488,9000000,62568000,57474376

%N T(n,k) = Number of (n+1) X (k+1) 0..2 arrays with every 2 X 2 subblock sum differing from its neighbors horizontally, vertically, diagonally and antidiagonally.

%C Table starts

%C .........81.........558.........3844........26784.......186624......1296000

%C ........558........3600........23546.......167832......1184928......8243488

%C .......3844.......23546.......140536......1029888......7431136.....52784020

%C ......26784......167832......1029888......8153960.....61286116....465921688

%C .....186624.....1184928......7431136.....61286116....486417836...4028778100

%C ....1296000.....8243488.....52784020....465921688...4028778100..36312321352

%C ....9000000....57474376....376392916...3524494796..32585471254.317292582684

%C ...62568000...402165016...2705942488..26879615668.268974784168

%C ..434972736..2811886280..19441124904.204459584768

%C .3022785216.19648735840.139538118106

%H R. H. Hardin, <a href="/A253294/b253294.txt">Table of n, a(n) for n = 1..84</a>

%F Empirical for column k:

%F k=1: [linear recurrence of order 9];

%F k=2: [order 40].

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

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

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

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

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

%Y Column 1 is A185771.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Dec 29 2014