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Decimal expansion of limit of the real part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.
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%I #13 Oct 23 2014 10:17:19

%S 5,2,9,0,8,5,5,1,3,6,3,5,7,4,6,1,2,5,1,6,0,9,9,0,5,2,3,7,9,0,2,2,5,2,

%T 1,0,6,1,9,3,6,5,0,4,9,8,3,8,9,0,9,7,4,3,1,4,0,7,7,1,1,7,6,3,2,0,2,3,

%U 9,8,1,1,5,7,9,1,8,9,4,6,2,7,7,1,1,4

%N Decimal expansion of limit of the real part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.

%C The analogous limit of f(1,n)/f(1,n+1) is the golden ratio (A001622).

%C Differs from A248749 only in the first digit. - _R. J. Mathar_, Oct 23 2014

%F Equals (sqrt(2+sqrt(5))-1)/2. - _Vaclav Kotesovec_, Oct 19 2014

%e limit = 0.52908551363574612516099052379022521061936504...

%e Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i.

%e n f(n,x) Re(q(c,n)) Im(q(c,n)))

%e 1 1 1/2 1/2

%e 2 x 3/5 1/5

%e 3 1 + x^2 1/2 1/4

%e 4 2x + x^3 8/15 4/15

%e 5 1 + 3 x^2 + x^4 69/130 33/130

%e Re(q(11,1+i) = 5021/9490 = 0.5290832...

%e Im(q(11,1+i) = 4879/18980 = 0.275060...

%p evalf((sqrt(2+sqrt(5))-1)/2, 120); # _Vaclav Kotesovec_, Oct 19 2014

%t z = 300; t = Table[Fibonacci[n, x]/Fibonacci[n + 1, x], {n, 1, z}];

%t u = t /. x -> 1 - I;

%t d1 = N[Re[u][[z]], 130]

%t d2 = N[Im[u][[z]], 130]

%t r1 = RealDigits[d1] (* A248751 *)

%t r2 = RealDigits[d2] (* A248752 *)

%Y Cf. A248750, A248752, A102426, A001622.

%K nonn,easy,cons

%O 0,1

%A _Clark Kimberling_, Oct 13 2014