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T(n,k)=Number of nXk binary arrays with top left value 1 and no two ones adjacent horizontally or vertically.
11

%I #7 Sep 01 2013 18:29:13

%S 1,1,1,2,2,2,3,5,5,3,5,12,21,12,5,8,29,72,72,29,8,13,70,268,382,268,

%T 70,13,21,169,963,2104,2104,963,169,21,34,408,3513,11449,17578,11449,

%U 3513,408,34,55,985,12732,62546,143072,143072,62546,12732,985,55,89,2378

%N T(n,k)=Number of nXk binary arrays with top left value 1 and no two ones adjacent horizontally or vertically.

%C Table starts

%C ..1...1.....2.......3........5..........8...........13............21

%C ..1...2.....5......12.......29.........70..........169...........408

%C ..2...5....21......72......268........963.........3513.........12732

%C ..3..12....72.....382.....2104......11449........62546........341249

%C ..5..29...268....2104....17578.....143072......1177709.......9646285

%C ..8..70...963...11449...143072....1755243.....21683149.....267157140

%C .13.169..3513...62546..1177709...21683149....402968942....7458864720

%C .21.408.12732..341249..9646285..267157140...7458864720..207573951234

%C .34.985.46274.1862631.79185086.3294926929.138305554175.5784184947686

%H R. H. Hardin, <a href="/A228390/b228390.txt">Table of n, a(n) for n = 1..1860</a>

%F Empirical for column k:

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

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

%F k=3: a(n) = 2*a(n-1) +6*a(n-2) -a(n-4)

%F k=4: a(n) = 4*a(n-1) +9*a(n-2) -5*a(n-3) -4*a(n-4) +a(n-5)

%F k=5: [order 9]

%F k=6: [order 11]

%F k=7: [order 21]

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 1 is A000045

%Y Column 2 is A000129

%K nonn,tabl

%O 1,4

%A _R. H. Hardin_ Aug 21 2013