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T(n,k)=Number of nXk 0..1 arrays with no element equal to the same number of vertical neighbors as horizontal neighbors, with new values 0..1 introduced in row major order
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%I #4 Apr 09 2014 19:17:26

%S 0,1,1,1,2,1,2,2,2,2,3,3,6,3,3,5,6,8,8,6,5,8,10,23,18,23,10,8,13,21,

%T 60,61,61,60,21,13,21,42,149,168,232,168,149,42,21,34,86,396,526,953,

%U 953,526,396,86,34,55,179,1050,1643,4343,5304,4343,1643,1050,179,55,89,370

%N T(n,k)=Number of nXk 0..1 arrays with no element equal to the same number of vertical neighbors as horizontal neighbors, with new values 0..1 introduced in row major order

%C Table starts

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

%C ..1...2....2.....3......6......10........21.........42..........86..........179

%C ..1...2....6.....8.....23......60.......149........396........1050.........2814

%C ..2...3....8....18.....61.....168.......526.......1643........5524........18762

%C ..3...6...23....61....232.....953......4343......19458.......90165.......421048

%C ..5..10...60...168....953....5304.....29481.....168320......990468......5920658

%C ..8..21..149...526...4343...29481....227270....1748201....14230080....116070258

%C .13..42..396..1643..19458..168320...1748201...18030130...191002776...2052931147

%C .21..86.1050..5524..90165..990468..14230080..191002776..2764522654..39961388170

%C .34.179.2814.18762.421048.5920658.116070258.2052931147.39961388170.770199142784

%H R. H. Hardin, <a href="/A240649/b240649.txt">Table of n, a(n) for n = 1..220</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) -a(n-3) -2*a(n-4) +a(n-5)

%F k=3: [order 20]

%F k=4: [order 48]

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 1 is A000045(n-1)

%Y Column 2 is A240513(n-2)

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Apr 09 2014