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%I #5 Mar 31 2012 12:36:19
%S 21,83,30,205,129,51,419,362,243,88,713,839,701,562,128,1169,1638,
%T 1715,2049,903,197,1709,2923,3343,5879,3442,1754,292,2471,4834,5947,
%U 13342,10259,8223,2937,457,3381,7605,9789,26982,23790,28628,14492,5790,688
%N T(n,m)=Number of (n+1)X3 0..m arrays with every 2X2 subblock commuting with each of its horizontal and vertical 2X2 subblock neighbors
%C Table starts
%C ...21....83....205....419.....713....1169.....1709.....2471......3381......4543
%C ...30...129....362....839....1638....2923.....4834.....7605.....11434.....16493
%C ...51...243....701...1715....3343....5947.....9789....15499.....23561.....33877
%C ...88...562...2049...5879...13342...26982....49867....87001....143918....224458
%C ..128...903...3442..10259...23790...48875....91096...160377....267770....420837
%C ..197..1754...8223..28628...77358..181792...383152...749237...1371936...2357920
%C ..292..2937..14492..53045..146406..349085...740408..1460393...2698956...4662219
%C ..457..5790..35081.150208..481171.1308972..3134584..6868867..13933702..26279979
%C ..688..9993..64280.289063..945714.2606517..6273818.13842067..28288458..53533915
%C .1058.19236.150785.792360.3006488.9456818.25715484.63104546.141749332.293291390
%H R. H. Hardin, <a href="/A190023/b190023.txt">Table of n, a(n) for n = 1..2450</a>
%F Empirical for column m: a(n) = 2*a(n-1) +(m-1)*a(n-2) -2*m*a(n-3) +2*m*a(n-4) -m*a(n-5) for n>(7+2*A072649(m)), except for n>8 for m=1.
%F Empirical more generally for column m of T(n,m,k) for (n+1)X(k+1) 0..m arrays: a(n) = 2*a(n-1) +(m-1)*a(n-2) -2*m*a(n-3) +2*m*a(n-4) -m*a(n-5) for n>(k+5+2*A072649(m)), except n>(k+4+2*A072649(m)) for m=1 and k>1
%e Some solutions for n=4 m=4
%e ..3..0..0....0..4..0....3..3..3....3..3..0....0..1..3....2..3..0....4..2..0
%e ..3..0..0....3..0..4....0..0..0....0..0..3....0..0..1....0..0..3....3..0..2
%e ..0..3..0....0..3..0....0..0..0....0..0..0....0..0..0....0..0..0....0..3..0
%e ..2..0..3....1..0..3....3..0..0....0..0..0....1..0..0....2..0..0....3..0..3
%e ..0..2..4....0..1..0....4..3..0....2..2..2....4..1..0....0..2..1....0..3..1
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_ May 04 2011