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%I #8 Oct 19 2014 15:43:35
%S 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 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 3,9,8,1,1,5,7,9,1,8,9,4,6,2,7,7,1,1
%N 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 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 Equals (1+sqrt(2+sqrt(5)))/2. - _Vaclav Kotesovec_, Oct 19 2014
%e limit = 1.52908551363574612516099052379022521061936504983890974314077117
%e 63202398115791894627711485520734841970516965799404...
%e n f(n,x) Re(f(n,1+i)) Im(f(n,1+i))
%e 1 1 1 0
%e 2 1 + x 2 1
%e 3 (1 + x + x^2)/ (1 + x) 7/5 4/5
%e 4 (1 + 2x + x^2 + x^3)/(1 + x + x^2) 20/13 9/13
%e Re(f(11,1+i) = 815/533 = 1.529162...
%e Im(f(11,1+i) = 396/533 = 0.742964...
%p evalf((1+sqrt(2+sqrt(5)))/2, 120); # _Vaclav Kotesovec_, Oct 19 2014
%t $RecursionLimit = Infinity; $MaxExtraPrecision = Infinity;
%t 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 {N[Re[u], 12], N[Im[u], 12]}
%t t = Table[Factor[f[x, n]], {n, 1, 300}]; u = t /. x -> I + 1;
%t r1 = N[Re[u][[300]], 130]
%t r2 = N[Im[u][[300]], 130]
%t d1 = RealDigits[r1] (* A248749 *)
%t d2 = RealDigits[r2] (* A248750 *)
%Y Cf. A248750, A046854.
%K nonn,easy,cons
%O 1,2
%A _Clark Kimberling_, Oct 13 2014
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