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T(n,k)=Number of nXk 0..1 arrays with every element unequal to 2 or 3 horizontally or vertically adjacent elements, with upper left element zero.
7

%I #4 Apr 27 2018 10:17:03

%S 0,0,0,0,1,0,0,1,1,0,0,2,0,2,0,0,3,1,1,3,0,0,5,0,4,0,5,0,0,8,3,8,8,3,

%T 8,0,0,13,2,22,20,22,2,13,0,0,21,9,47,66,66,47,9,21,0,0,34,12,123,207,

%U 394,207,123,12,34,0,0,55,33,302,726,1417,1417,726,302,33,55,0,0,89,56,784

%N T(n,k)=Number of nXk 0..1 arrays with every element unequal to 2 or 3 horizontally or vertically adjacent elements, with upper left element zero.

%C Table starts

%C .0..0..0...0....0.....0......0.......0........0.........0..........0

%C .0..1..1...2....3.....5......8......13.......21........34.........55

%C .0..1..0...1....0.....3......2.......9.......12........33.........56

%C .0..2..1...4....8....22.....47.....123......302.......784.......1961

%C .0..3..0...8...20....66....207.....726.....2461......8538......29072

%C .0..5..3..22...66...394...1417....7706....33066....162051.....742361

%C .0..8..2..47..207..1417...7538...53989...309136...2041378...12561807

%C .0.13..9.123..726..7706..53989..501126..4171831..35313591..303281789

%C .0.21.12.302.2461.33066.309136.4171831.43679228.535178476.6028110834

%H R. H. Hardin, <a href="/A303636/b303636.txt">Table of n, a(n) for n = 1..264</a>

%F Empirical for column k:

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

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

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

%F k=4: [order 16]

%F k=5: [order 39]

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 2 is A000045(n-1).

%K nonn,tabl

%O 1,12

%A _R. H. Hardin_, Apr 27 2018