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Number of 8 X (n+1) 0..1 arrays with every 2 X 2 subblock having the same number of equal edges as its horizontal neighbors and a different number from its vertical neighbors, and new values 0..1 introduced in row major order.
1

%I #11 Jun 28 2018 08:25:52

%S 216,504,1206,3096,7992,20862,54504,142632,373302,977256,2558376,

%T 6697854,17535096,45907416,120187062,314653752,823774104,2156668542,

%U 5646231432,14782025736,38699845686,101317511304,265252688136,694440553086

%N Number of 8 X (n+1) 0..1 arrays with every 2 X 2 subblock having the same number of equal edges as its horizontal neighbors and a different number from its vertical neighbors, and new values 0..1 introduced in row major order.

%C Row 7 of A208085.

%H R. H. Hardin, <a href="/A208089/b208089.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).

%F Conjectures from _Colin Barker_, Jun 28 2018: (Start)

%F G.f.: 18*x*(12 - 8*x - 17*x^2 + 7*x^3) / ((1 - x)*(1 + x)*(1 - 3*x + x^2)).

%F a(n) = (9/5)*2^(1-n)*(2^n*(15+2*(-1)^n) + (9-4*sqrt(5))*(3-sqrt(5))^n + (3+sqrt(5))^n*(9+4*sqrt(5))).

%F (End)

%e Some solutions for n=4:

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

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

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

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

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

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

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

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

%Y Cf. A208085.

%K nonn

%O 1,1

%A _R. H. Hardin_, Feb 23 2012