%I #8 Apr 20 2013 08:17:19

%S 8,20,20,56,84,56,164,376,376,164,488,1708,2606,1708,488,1460,7784,

%T 18152,18152,7784,1460,4376,35500,126536,193664,126536,35500,4376,

%U 13124,161928,882182,2068148,2068148,882182,161928,13124,39368,738636,6150512

%N T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with every 2X2 subblock having the same number of equal edges, and new values 0..1 introduced in row major order.

%C Table starts

%C .....8.....20.......56........164..........488..........1460............4376

%C ....20.....84......376.......1708.........7784.........35500..........161928

%C ....56....376.....2606......18152.......126536........882182.........6150512

%C ...164...1708....18152.....193664......2068148......22091516.......235994088

%C ...488...7784...126536....2068148.....33865634.....554916092......9094954742

%C ..1460..35500...882182...22091516....554916092...13956665238....351210375464

%C ..4376.161928..6150512..235994088...9094954742..351210375464..13574876544398

%C .13124.738636.42881096.2521075824.149077423220.8839958693704.524918733085720

%H R. H. Hardin, <a href="/A205318/b205318.txt">Table of n, a(n) for n = 1..420</a>

%F Empirical for column k:

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

%F k=2: a(n) = 6*a(n-1) -7*a(n-2) +2*a(n-3)

%F k=3: a(n) = 10*a(n-1) -24*a(n-2) +21*a(n-3) -6*a(n-4)

%F k=4: a(n) = 17*a(n-1) -81*a(n-2) +157*a(n-3) -140*a(n-4) +56*a(n-5) -8*a(n-6)

%F k=5: a(n) = 31*a(n-1) -321*a(n-2) +1569*a(n-3) -4179*a(n-4) +6420*a(n-5) -5671*a(n-6) +2668*a(n-7) -516*a(n-8)

%F k=6: (order 14 recurrence)

%F k=7: (order 20 recurrence)

%e Some solutions for n=4 k=3

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

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

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

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

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

%Y Column 1 is A115099.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_ Jan 25 2012