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T(n,k)=Number of nXk integer arrays with each element equal to the number of horizontal and vertical neighbors differing from itself by exactly one.
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%I #4 Dec 20 2015 17:31:21

%S 1,1,1,2,1,2,1,4,4,1,1,1,4,1,1,2,1,6,6,1,2,1,4,5,5,5,4,1,1,1,12,7,7,

%T 12,1,1,2,1,10,26,29,26,10,1,2,1,4,16,23,29,29,23,16,4,1,1,1,21,35,55,

%U 78,55,35,21,1,1,2,1,31,64,115,146,146,115,64,31,1,2,1,4,42,99,221,364,466,364

%N T(n,k)=Number of nXk integer arrays with each element equal to the number of horizontal and vertical neighbors differing from itself by exactly one.

%C Table starts

%C .1.1..2..1...1....2....1.....1......2......1.......1........2........1

%C .1.1..4..1...1....4....1.....1......4......1.......1........4........1

%C .2.4..4..6...5...12...10....16.....21.....31......42.......66.......89

%C .1.1..6..5...7...26...23....35.....64.....99.....161......276......431

%C .1.1..5..7..29...29...55...115....221....443.....851.....1659.....3318

%C .2.4.12.26..29...78..146...364....783...1773....4170.....9548....21795

%C .1.1.10.23..55..146..466..1155...3364...9229...25566....70308...196647

%C .1.1.16.35.115..364.1155..3907..12510..41683..136701...446548..1484311

%C .2.4.21.64.221..783.3364.12510..52773.196401..781997..3116201.12334789

%C .1.1.31.99.443.1773.9229.41683.196401.925875.4351555.20635583.96976290

%H R. H. Hardin, <a href="/A266081/b266081.txt">Table of n, a(n) for n = 1..264</a>

%F Empirical for column k:

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

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

%F k=3: a(n) = a(n-1) +a(n-3) +a(n-6) -a(n-7) -a(n-9)

%F k=4: [order 63]

%e All solutions for n=4 k=4

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

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

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

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

%K nonn,tabl

%O 1,4

%A _R. H. Hardin_, Dec 20 2015