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A248751 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.

%I A248751 #18 Feb 27 2025 05:05:43
%S A248751 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 A248751 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 A248751 9,8,1,1,5,7,9,1,8,9,4,6,2,7,7,1,1,4
%N A248751 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 A248751 The analogous limit of f(1,n)/f(1,n+1) is the golden ratio (A001622).
%C A248751 Differs from A248749 only in the first digit. - _R. J. Mathar_, Oct 23 2014
%H A248751 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F A248751 Equals (sqrt(2+sqrt(5))-1)/2. - _Vaclav Kotesovec_, Oct 19 2014
%e A248751 limit = 0.52908551363574612516099052379022521061936504...
%e A248751 Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i.
%e A248751 n   f(x,n)            Re(q(c,n))   Im(q(c,n))
%e A248751 1   1                 1/2          1/2
%e A248751 2   x                 3/5          1/5
%e A248751 3   1 + x^2           1/2          1/4
%e A248751 4   2*x + x^3         8/15         4/15
%e A248751 5   1 + 3*x^2 + x^4   69/130       33/130
%e A248751 Re(q(1-i,11)) = 5021/9490 = 0.5290832...
%e A248751 Im(q(1-i,11)) = 4879/18980 = 0.257060...
%p A248751 evalf((sqrt(2+sqrt(5))-1)/2, 120); # _Vaclav Kotesovec_, Oct 19 2014
%t A248751 z = 300; t = Table[Fibonacci[n, x]/Fibonacci[n + 1, x], {n, 1, z}];
%t A248751 u = t /. x -> 1 - I;
%t A248751 d1 = N[Re[u][[z]], 130]
%t A248751 d2 = N[Im[u][[z]], 130]
%t A248751 r1 = RealDigits[d1]  (* A248751 *)
%t A248751 r2 = RealDigits[d2]  (* A248752 *)
%o A248751 (PARI) polrootsreal(4*x^4+8*x^3+2*x^2-2*x-1)[2] \\ _Charles R Greathouse IV_, Nov 26 2024
%Y A248751 Cf. A248750, A248752, A102426, A001622.
%K A248751 nonn,easy,cons
%O A248751 0,1
%A A248751 _Clark Kimberling_, Oct 13 2014