%I #6 Oct 01 2013 13:00:31

%S 2,3,4,8,4,0,7,4,7,0,2,7,9,2,3,0,1,7,7,5,3,9,4,2,1,0,6,1,9,7,5,6,8,4,

%T 4,6,5,9,9,4,5,9,1,3,4,1,9,4,4,3,6,3,7,9,2,4,0,6,8,6,0,9,3,9,3,3,8,1,

%U 8,6,8,6,5,2,7,8,2,1,1,7,2,8,8,8,2,2,5,8,7,0,0,9,6,9,0,7,5,5,1,7,1,4,2,0,3,9,6,1,2,2,5,9,5,9,6,7

%N Decimal expansion of L(e), the limit of iterations of continued fraction transforms of e.

%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), ... ].

%C Thus, f(e) = 2.9991459..., f(f(e)) = 2.3690966..., f(f(f(e))) = 2.3483570.... 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(e) = 2.34840747..., .

%C Conjecture: if x is an irrational number between 2 and 3, then L(x) = L(e).

%e L(e) = 2.348407470279230177539421061975684465994591341944...

%t $MaxExtraPrecision = Infinity;

%t z = 600; x[0] = E; c[0] = Convergents[x[0], z];

%t x[n_] := N[FromContinuedFraction[c[n - 1]], 80];

%t c[n_] := Convergents[x[n]];

%t Table[x[n], {n, 1, 20}] (* f(e), f(f(e)), ... *)

%t RealDigits[x[1]] (* f(e), A229594 *)

%t Numerator[c[1]] (* A229595 *)

%t Denominator[c[1]] (* A229596 *)

%Y Cf. A229594, A229595, A229596.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Sep 26 2013