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T(n,k) is the number of horizontal, vertical and diagonal neighbor colorings of the odd squares of an n X k array with new integer colors introduced in row major order.
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%I #10 Oct 30 2024 08:06:41

%S 1,1,1,1,2,1,1,2,2,1,1,5,4,5,1,2,12,29,29,12,2,2,42,97,371,97,42,2,5,

%T 136,1534,6150,6150,1534,136,5,5,602,7050,167130,146327,167130,7050,

%U 602,5,15,2440,188835,5219070,30312074,30312074,5219070,188835,2440,15,15,12840

%N T(n,k) is the number of horizontal, vertical and diagonal neighbor colorings of the odd squares of an n X k array with new integer colors introduced in row major order.

%H R. H. Hardin, <a href="/A216031/b216031.txt">Table of n, a(n) for n = 1..84</a>

%e Table starts:

%e ..1.....1........1...........1............1.............2.............2

%e ..1.....2........2...........5...........12............42...........136

%e ..1.....2........4..........29...........97..........1534..........7050

%e ..1.....5.......29.........371.........6150........167130.......5219070

%e ..1....12.......97........6150.......146327......30312074....1355988481

%e ..2....42.....1534......167130.....30312074...10311562327.4761748786964

%e ..2...136.....7050.....5219070...1355988481.4761748786964

%e ..5...602...188835...233834847.609400419815

%e ..5..2440..1084180.11412814314

%e .15.12840.42756208

%e .15.61912

%e .52

%e Some solutions for n=4 and k=4:

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

%e ..2..x..1..x....2..x..3..x....0..x..2..x....0..x..1..x....0..x..1..x

%e ..x..1..x..2....x..1..x..4....x..3..x..0....x..2..x..0....x..1..x..0

%e ..0..x..2..x....3..x..2..x....4..x..0..x....3..x..4..x....1..x..2..x

%Y Column 1 is A000110(floor((n-2)/2)) for n>1.

%Y Even squares: A215959.

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Aug 30 2012