%I #46 Jun 05 2021 01:55:11
%S 1,1,2,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,
%T 8362,13530,21892,35422,57314,92736,150050,242786,392836,635622,
%U 1028458,1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634,78176338
%N Number of n X 2 binary arrays with all 1's connected, all corners 1, and no 1 having more than two 1's adjacent.
%C Same recurrence for A163695.
%C Same recurrence for A163714.
%C Appears to coincide with diagonal sums of A072405. - _Paul Barry_, Aug 10 2009
%C From _Gary W. Adamson_, Sep 15 2016: (Start)
%C Let the sequence prefaced with a 1: (1, 1, 1, 2, 2, 4, 6, ...) equate to r(x). Then (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) = the Fibonacci sequence, (1, 1, 2, 3, 5, ...). Let M = the following production matrix:
%C 1, 0, 0, 0, 0, ...
%C 1, 0, 0, 0, 0, ...
%C 1, 1, 0, 0, 0, ...
%C 2, 1, 0, 0, 0, ...
%C 2, 1, 1, 0, 0, ...
%C 4, 2, 1, 0, 0, ...
%C 6, 2, 1, 1, 0, ...
%C ...
%C Limit of the matrix power M^k as k->infinity results in a single column vector equal to the Fibonacci sequence. (End)
%C Apparently a(n) = A128588(n-2) for n > 3. - _Georg Fischer_, Oct 14 2018
%H R. H. Hardin, <a href="/A163733/b163733.txt">Table of n, a(n) for n=1..100</a>
%F Empirical: a(n) = a(n-1) + a(n-2) for n >= 5.
%F G.f.: (1-x^3)/(1-x-x^2) (conjecture). - _Paul Barry_, Aug 10 2009
%F a(n) = round(phi^(k-1)) - round(phi^(k-1)/sqrt(5)), phi = (1 + sqrt(5))/2 (conjecture). - _Federico Provvedi_, Mar 26 2013
%F G.f.: 1 + 2*x - x*Q(0), where Q(k) = 1 + x^2 - (2*k+1)*x + x*(2*k-1 - x)/Q(k+1); (conjecture), (continued fraction). - _Sergei N. Gladkovskii_, Oct 05 2013
%F G.f.: If prefaced with a 1, (1, 1, 1, 2, 2, 4, ...): (1 - x^2 - x^4)/(1 - x - x^2); where the modified sequence satisfies A(x)/A(x^2), A(x) is the Fibonacci sequence. - _Gary W. Adamson_, Sep 15 2016
%e All solutions for n=8:
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1
%e 0 1 1 0 1 0 1 0 1 1 1 0 0 1 0 1 1 1 0 1
%e 0 1 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1
%e 0 1 1 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0
%e 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 0
%e 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e ------
%e 1 1 1 1 1 1 1 1 1 1 1 1
%e 0 1 0 1 0 1 1 0 1 0 1 0
%e 1 1 1 1 0 1 1 0 1 1 1 1
%e 1 0 1 0 1 1 1 1 0 1 0 1
%e 1 1 1 0 1 0 0 1 0 1 1 1
%e 0 1 1 1 1 1 1 1 1 1 1 0
%e 0 1 0 1 0 1 1 0 1 0 1 0
%e 1 1 1 1 1 1 1 1 1 1 1 1
%t Join[{1, 1}, Table[2*Fibonacci[n], {n, 70}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 10 2012 *)
%t Table[Round[GoldenRatio^(k-1)] - Round[GoldenRatio^(k-1)/Sqrt[5]], {k, 1, 70}] (* _Federico Provvedi_, Mar 26 2013 *)
%Y Cf. A118658, A055389, A006355, A006355, A169985, A000045.
%Y Cf. A128588. - _Georg Fischer_, Oct 14 2018
%K nonn
%O 1,3
%A _R. H. Hardin_, Aug 03 2009