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A248749 Decimal expansion of limit of the real part of f(1+i,n), where f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1).

%I A248749 #12 Feb 27 2025 05:05:35
%S A248749 1,5,2,9,0,8,5,5,1,3,6,3,5,7,4,6,1,2,5,1,6,0,9,9,0,5,2,3,7,9,0,2,2,5,
%T A248749 2,1,0,6,1,9,3,6,5,0,4,9,8,3,8,9,0,9,7,4,3,1,4,0,7,7,1,1,7,6,3,2,0,2,
%U A248749 3,9,8,1,1,5,7,9,1,8,9,4,6,2,7,7,1,1
%N A248749 Decimal expansion of limit of the real part of f(1+i,n), where f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1).
%C A248749 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.
%F A248749 Equals (1+sqrt(2+sqrt(5)))/2. - _Vaclav Kotesovec_, Oct 19 2014
%e A248749 limit = 1.52908551363574612516099052379022521061936504983890974314077117...
%e A248749 n   f(x,n)                                 Re(f(1+i,n))  Im(f(1+i,n))
%e A248749 0   1                                      1             0
%e A248749 1   1 + x                                  2             1
%e A248749 2   (1 + x + x^2)/(1 + x)                  7/5           4/5
%e A248749 3   (1 + 2*x + x^2 + x^3)/(1 + x + x^2)    20/13         9/13
%e A248749 Re(f(1+i,10)) = 815/533 = 1.529080...
%e A248749 Im(f(1+i,10)) = 396/533 = 0.742964...
%p A248749 evalf((1+sqrt(2+sqrt(5)))/2, 120); # _Vaclav Kotesovec_, Oct 19 2014
%t A248749 $RecursionLimit = Infinity; $MaxExtraPrecision = Infinity;
%t A248749 f[x_, n_] := x + 1/f[x, n - 1]; f[x_, 1] = 1; t = Table[Factor[f[x, n]], {n, 1, 12}]; u = t /. x -> I + 1; {Re[u], Im[u]}
%t A248749 {N[Re[u], 12], N[Im[u], 12]}
%t A248749 t = Table[Factor[f[x, n]], {n, 1, 300}]; u = t /. x -> I + 1;
%t A248749 r1 = N[Re[u][[300]], 130]
%t A248749 r2 = N[Im[u][[300]], 130]
%t A248749 d1 = RealDigits[r1]  (* A248749 *)
%t A248749 d2 = RealDigits[r2]  (* A248750 *)
%Y A248749 Cf. A248750, A046854.
%K A248749 nonn,easy,cons
%O A248749 1,2
%A A248749 _Clark Kimberling_, Oct 13 2014