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%I #13 Mar 02 2018 03:56:14
%S 2,5,7,0,6,5,8,6,4,1,2,1,6,7,7,1,6,0,9,0,8,5,6,8,0,6,2,0,5,2,7,3,3,7,
%T 1,8,9,0,3,7,5,7,0,0,7,9,9,8,8,1,3,4,9,4,5,2,4,1,3,0,7,9,3,7,8,0,9,4,
%U 2,2,3,6,0,4,3,1,2,1,4,5,0,9,4,0,7,6
%N Decimal expansion of limit of the imaginary 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),
%F Equals (1-sqrt(sqrt(5)-2))/2. - _Vaclav Kotesovec_, Oct 19 2014
%F From _Wolfdieter Lang_, Mar 02 2018: (Start)
%F Equals (1 - (2 - phi)*sqrt(phi))/2, with phi = A001622.
%F Equals (1/10)*y*(1 - (1/50)*y^2) with y = A300070. (End)
%e limit = 0.2570658641216771609085680620527337189037570...
%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((1-sqrt(sqrt(5)-2))/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, A248751, A102426, A001622, A300070.
%K nonn,easy,cons
%O 0,1
%A _Clark Kimberling_, Oct 13 2014
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