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%I #7 Oct 06 2013 13:55:24
%S 3,2,7,6,5,0,3,3,8,5,0,1,4,4,2,4,4,6,3,1,3,8,6,9,7,2,3,5,0,0,1,9,1,0,
%T 2,1,8,3,6,4,2,5,5,3,8,4,1,6,8,0,6,5,4,0,9,1,7,4,2,2,2,0,8,4,8,0,1,7,
%U 5,5,0,4,9,9,5,1,6,2,9,0,2,8,9,5,4,5
%N Decimal expansion of L(Pi), the limit of iterations of continued fraction transforms of Pi.
%C The function f defined at A229350 is here called the continued fraction transform; specifically, to define f(x), start with x > 0: let p(i)/q(i), for i >=0, be the convergents to x; then f(x) is the number [p(0)/q(0), p(1)/q(1), p(2)/q(2), ... ]. Thus, f(Pi) = 3.291191..., f(f(Pi)) = 3.276718..., f(f(f(Pi))) =3.276503 ...; let L(x) = lim(f(n,x)), where f(0,x) = x, f(1,x) = f(x), and f(n,x) = f(f(n-1,x)). Then L(Pi) =3.276503 ..., as in A228993.
%C Conjecture: if x is an irrational number between 3 and 4, then L(x) = L(Pi).
%e L(Pi) = 3.2765033850144244631386972350019102183642553841680654...
%t $MaxExtraPrecision = Infinity;
%t z = 600; x[0] = Pi; c[0] = Convergents[x[0], z]; x[n_] := N[FromContinuedFraction[c[n - 1]], 80]; c[n_] := Convergents[x[n]]; Table[x[n], {n, 1, 20}] (* A228492, f(Pi), f(f(Pi)), ... *)
%t t1 = RealDigits[x[1]] (* f(Pi), A228493 *)
%t t2 = Numerator[c[1]] (* A228992 *)
%t t3 = Denominator[c[1]] (* A228993 *)
%Y Cf. A228492, A229597 (L(e)).
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Oct 01 2013
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