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Number of n X 4 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.
1

%I #8 Jul 06 2018 06:04:49

%S 8,100,1268,16084,204020,2587924,32826932,416398420,5281871732,

%T 66998738836,849856117940,10780134577876,136742325040628,

%U 1734529687216660,22001916633654068,279086797488636244

%N Number of n X 4 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.

%C Column 4 of A208709.

%H R. H. Hardin, <a href="/A208705/b208705.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 13*a(n-1) - 4*a(n-2).

%F Conjectures from _Colin Barker_, Jul 06 2018: (Start)

%F G.f.: 4*x*(2 - x) / (1 - 13*x + 4*x^2).

%F a(n) = (2^(-1-n)*((13-3*sqrt(17))^n*(-1+sqrt(17)) + (1+sqrt(17))*(13+3*sqrt(17))^n)) / sqrt(17).

%F (End)

%e Some solutions for n=4:

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

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

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

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

%Y Cf. A208709

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 01 2012