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A248750 Decimal expansion of limit of the imaginary part of f(1+i,n), where f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1).

%I A248750 #25 Feb 27 2025 05:05:39
%S A248750 7,4,2,9,3,4,1,3,5,8,7,8,3,2,2,8,3,9,0,9,1,4,3,1,9,3,7,9,4,7,2,6,6,2,
%T A248750 8,1,0,9,6,2,4,2,9,9,2,0,0,1,1,8,6,5,0,5,4,7,5,8,6,9,2,0,6,2,1,9,0,5,
%U A248750 7,7,6,3,9,5,6,8,7,8,5,4,9,0,5,9,2,3
%N A248750 Decimal expansion of limit of the imaginary part of f(1+i,n), where f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1).
%C A248750 See A046854 for a triangle of coefficients of the numerators and denominators of f(x,n). Note that the limit of f(1,n) is the golden ratio.
%H A248750 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A248750 Equals (1+sqrt(sqrt(5)-2))/2. - _Vaclav Kotesovec_, Oct 19 2014
%F A248750 From _Wolfdieter Lang_, Mar 02 2018: (Start)
%F A248750 Equals (1 + (2 - phi)*sqrt(phi))/2, with phi = A001622.
%F A248750 Equals (1/10)*y*(1 - (1/50)*y^2) with y = -A300072. (End)
%e A248750 0.742934135878322839091431937947266281096242992001186505475869206219...
%e A248750 n   f(x,n)                                 Re(f(1+i,n))  Im(f(1+i,n))
%e A248750 0   1                                      1             0
%e A248750 1   1 + x                                  2             1
%e A248750 2   (1 + x + x^2)/(1 + x)                  7/5           4/5
%e A248750 3   (1 + 2*x + x^2 + x^3)/(1 + x + x^2)    20/13         9/13
%e A248750 Re(f(1+i,10)) = 815/533 = 1.529080...
%e A248750 Im(f(1+i,10)) = 396/533 = 0.742964...
%p A248750 evalf((1+sqrt(sqrt(5)-2))/2, 120); # _Vaclav Kotesovec_, Oct 19 2014
%t A248750 $RecursionLimit = Infinity; $MaxExtraPrecision = Infinity;
%t A248750 z = 20; (* For more accuracy, increase z *)
%t A248750 f[x_, n_] := x + 1/f[x, n - 1];
%t A248750 f[x_, 1] = 1; t = Table[Factor[f[x, n]], {n, 1, z}];
%t A248750 u = t /. x -> I + 1; t = Table[Factor[f[x, n]], {n, 1, z}]; u = t /. x -> I + 1;
%t A248750 r1 = N[Re[u][[z]], 130]
%t A248750 r2 = N[Im[u][[z]], 130]
%t A248750 d1 = RealDigits[r1]  (*A248749*)
%t A248750 d2 = RealDigits[r2]  (*A248750*)
%Y A248750 Cf. A248749, A046854, A248751, A248752, A001622. A300072.
%K A248750 nonn,easy,cons
%O A248750 0,1
%A A248750 _Clark Kimberling_, Oct 13 2014