cp's OEIS Frontend

A163704 Number of n X 2 binary arrays with all 1s connected, a path of 1s from left column to lower right corner, and no 1 having more than two 1s adjacent.

%I A163704 #11 Mar 26 2018 09:53:44
%S A163704 1,5,11,21,38,66,112,187,309,507,828,1348,2190,3553,5759,9329,15106,
%T A163704 24454,39580,64055,103657,167735,271416,439176,710618,1149821,1860467,
%U A163704 3010317,4870814,7881162,12752008,20633203,33385245,54018483,87403764
%N A163704 Number of n X 2 binary arrays with all 1s connected, a path of 1s from left column to lower right corner, and no 1 having more than two 1s adjacent.
%H A163704 R. H. Hardin, <a href="/A163704/b163704.txt">Table of n, a(n) for n = 1..100</a>
%F A163704 Empirical: a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n >= 6.
%F A163704 Conjectures from _Colin Barker_, Mar 25 2018: (Start)
%F A163704 G.f.: x*(1 + 2*x - 2*x^2 - x^3 + x^4) / ((1 - x)^2*(1 - x - x^2)).
%F A163704 a(n) = -4 + (2^(-n)*((1-sqrt(5))^n*(-5+2*sqrt(5)) + (1+sqrt(5))^n*(5+2*sqrt(5)))) / sqrt(5) - n for n>1.
%F A163704 (End)
%e A163704 All solutions for n=3:
%e A163704   0 0   0 0   0 0   0 0   1 0   0 1   1 1   0 0   1 0   1 1   1 1
%e A163704   0 0   0 1   1 0   1 1   1 0   0 1   1 0   1 1   1 1   0 1   0 1
%e A163704   1 1   1 1   1 1   1 1   1 1   1 1   1 1   0 1   0 1   0 1   1 1
%Y A163704 Cf. A023548. - _R. J. Mathar_, Aug 06 2009
%K A163704 nonn
%O A163704 1,2
%A A163704 _R. H. Hardin_, Aug 03 2009