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A248752 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.

%I A248752 #16 Feb 27 2025 05:05:46
%S A248752 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 A248752 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 A248752 2,2,3,6,0,4,3,1,2,1,4,5,0,9,4,0,7,6
%N A248752 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 A248752 The analogous limit of f(1,n)/f(1,n+1) is the golden ratio (A001622).
%F A248752 Equals (1-sqrt(sqrt(5)-2))/2. - _Vaclav Kotesovec_, Oct 19 2014
%F A248752 From _Wolfdieter Lang_, Mar 02 2018: (Start)
%F A248752 Equals (1 - (2 - phi)*sqrt(phi))/2, with phi = A001622.
%F A248752 Equals (1/10)*y*(1 - (1/50)*y^2) with y = A300070. (End)
%e A248752 limit = 0.2570658641216771609085680620527337189037570...
%e A248752 Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i.
%e A248752 n   f(x,n)            Re(q(c,n))   Im(q(c,n))
%e A248752 1   1                 1/2          1/2
%e A248752 2   x                 3/5          1/5
%e A248752 3   1 + x^2           1/2          1/4
%e A248752 4   2*x + x^3         8/15         4/15
%e A248752 5   1 + 3*x^2 + x^4   69/130       33/130
%e A248752 Re(q(1-i,11)) = 5021/9490 = 0.5290832...
%e A248752 Im(q(1-i,11)) = 4879/18980 = 0.257060...
%p A248752 evalf((1-sqrt(sqrt(5)-2))/2, 120); # _Vaclav Kotesovec_, Oct 19 2014
%t A248752 z = 300; t = Table[Fibonacci[n, x]/Fibonacci[n + 1, x], {n, 1, z}];
%t A248752 u = t /. x -> 1 - I;
%t A248752 d1 = N[Re[u][[z]], 130]
%t A248752 d2 = N[Im[u][[z]], 130]
%t A248752 r1 = RealDigits[d1]  (* A248751 *)
%t A248752 r2 = RealDigits[d2]  (* A248752 *)
%Y A248752 Cf. A248750, A248751, A102426, A001622, A300070.
%K A248752 nonn,easy,cons
%O A248752 0,1
%A A248752 _Clark Kimberling_, Oct 13 2014