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A163714 Number of n X 2 binary arrays with all 1s connected, a path of 1s from top row to bottom row, and no 1 having more than two 1s adjacent.

%I A163714 #9 Feb 22 2018 06:17:07
%S A163714 3,7,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,8362,
%T A163714 13530,21892,35422,57314,92736,150050,242786,392836,635622,1028458,
%U A163714 1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634,78176338
%N A163714 Number of n X 2 binary arrays with all 1s connected, a path of 1s from top row to bottom row, and no 1 having more than two 1s adjacent.
%C A163714 Same recurrence for A163695.
%C A163714 Same recurrence for A163733.
%H A163714 R. H. Hardin, <a href="/A163714/b163714.txt">Table of n, a(n) for n=1..100</a>
%F A163714 Empirical: a(n) = a(n-1) + a(n-2) for n>=5.
%F A163714 Conjectures from _Colin Barker_, Feb 22 2018: (Start)
%F A163714 G.f.: x*(1 + x)*(3 + x - x^2) / (1 - x - x^2).
%F A163714 a(n) = (2^(-n)*((1-sqrt(5))^n*(-3+sqrt(5)) + (1+sqrt(5))^n*(3+sqrt(5)))) / sqrt(5) for n>2.
%F A163714 (End)
%e A163714 All solutions for n=4:
%e A163714 ...1.0...1.0...1.1...1.1...0.1...0.1...1.1...1.1...1.0...1.1...1.0...1.0...0.1
%e A163714 ...1.0...1.0...1.0...1.0...0.1...0.1...0.1...0.1...1.0...1.0...1.1...1.1...0.1
%e A163714 ...1.0...1.0...1.0...1.0...0.1...0.1...0.1...0.1...1.1...1.1...0.1...0.1...1.1
%e A163714 ...1.0...1.1...1.0...1.1...0.1...1.1...0.1...1.1...0.1...0.1...0.1...1.1...1.0
%e A163714 ------
%e A163714 ...1.1...0.1...0.1
%e A163714 ...0.1...1.1...1.1
%e A163714 ...1.1...1.0...1.0
%e A163714 ...1.0...1.0...1.1
%Y A163714 Cf. A090991, A078642, A047992. - _R. J. Mathar_, Aug 06 2009
%K A163714 nonn
%O A163714 1,1
%A A163714 _R. H. Hardin_, Aug 03 2009