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A188516 Number of nX2 binary arrays without the pattern 1 1 0 diagonally, vertically or horizontally.

%I A188516 #19 Jul 22 2025 11:14:47
%S A188516 4,16,49,144,400,1089,2916,7744,20449,53824,141376,370881,972196,
%T A188516 2547216,6671889,17472400,45751696,119793025,313644100,821166336,
%U A188516 2149898689,5628600576,14736017664,38579637889,101003196100,264430435984
%N A188516 Number of nX2 binary arrays without the pattern 1 1 0 diagonally, vertically or horizontally.
%C A188516 Column 2 of A188523
%H A188516 R. H. Hardin, <a href="/A188516/b188516.txt">Table of n, a(n) for n = 1..200</a>
%F A188516 Empirical: a(n)=4*a(n-1)-2*a(n-2)-6*a(n-3)+4*a(n-4)+2*a(n-5)-a(n-6).
%F A188516 Conjecture:  a(n) = (F(n+3) - 1)^2, where F = A000045 (Fibonacci numbers). - _Clark Kimberling_, Jun 21 2016
%F A188516 Assuming the conjecture, define b(1) = 1 and b(n) = a(n-1) for n > 1.   Then b(n) = Sum{F(i,j): (i=n and 1<=j<=n) or (j=n and 1<=i<=n)}, where F is the Fibonacci fusion array, A202453. - _Clark Kimberling_, Jun 21 2016
%F A188516 G.f. for (b(n)):  -x*(-1+x^3-2*x^2) / ( (x-1)*(1+x)*(x^2-3*x+1)*(x^2+x-1) ). - _R. J. Mathar_, Dec 20 2011
%F A188516 b(n) = -2*(-1)^n/5 - 2*Fibonacci(n+2) + Lucas(2*n+4)/5 + 1. - _Ehren Metcalfe_, Mar 26 2016
%e A188516 Some solutions for 3X2
%e A188516 ..0..1....0..1....0..0....0..0....1..0....0..1....1..0....0..1....0..0....0..1
%e A188516 ..0..0....0..0....0..0....0..1....1..1....1..0....0..1....0..1....1..0....1..0
%e A188516 ..1..1....0..0....0..1....1..0....1..1....0..0....1..0....1..1....0..0....0..1
%K A188516 nonn
%O A188516 1,1
%A A188516 _R. H. Hardin_, Apr 02 2011