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A163695 Number of n X 2 binary arrays with all 1s connected, a path of 1s from top row to lower right corner, and no 1 having more than two 1s adjacent.

%I A163695 #16 Feb 20 2018 04:27:32
%S A163695 2,5,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,
%T A163695 15127,24476,39603,64079,103682,167761,271443,439204,710647,1149851,
%U A163695 1860498,3010349,4870847,7881196,12752043,20633239,33385282,54018521,87403803
%N A163695 Number of n X 2 binary arrays with all 1s connected, a path of 1s from top row to lower right corner, and no 1 having more than two 1s adjacent.
%H A163695 R. H. Hardin, <a href="/A163695/b163695.txt">Table of n, a(n) for n=1..100</a>
%H A163695 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A163695 a(n) = a(n-1) + a(n-2) for n>=5.
%F A163695 [The Transfer Matrix Method provides this recurrence. - _R. J. Mathar_, Aug 02 2017]
%F A163695 From _Colin Barker_, Feb 20 2018: (Start)
%F A163695 G.f.: x*(2 - x)*(1 + x)^2 / (1 - x - x^2).
%F A163695 a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-5+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5)))) / sqrt(5) for n>2.
%F A163695 (End)
%e A163695 All solutions for n=4:
%e A163695 ...0.1...0.1...1.1...1.1...1.0...1.1...1.0...1.1...1.0...1.0...0.1
%e A163695 ...0.1...0.1...0.1...0.1...1.0...1.0...1.0...1.0...1.1...1.1...1.1
%e A163695 ...0.1...0.1...0.1...0.1...1.1...1.1...1.0...1.0...0.1...0.1...1.0
%e A163695 ...0.1...1.1...0.1...1.1...0.1...0.1...1.1...1.1...0.1...1.1...1.1
%o A163695 (PARI) Vec(x*(2 - x)*(1 + x)^2 / (1 - x - x^2) + O(x^60)) \\ _Colin Barker_, Feb 20 2018
%Y A163695 It appears that A163714 and A163733 have the same recurrence as this sequence.
%Y A163695 Cf. A288219.
%K A163695 nonn,easy
%O A163695 1,1
%A A163695 _R. H. Hardin_, Aug 03 2009