OFFSET
1,1
COMMENTS
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(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..., .
Conjecture: if x is an irrational number between 2 and 3, then L(x) = L(e).
EXAMPLE
L(e) = 2.348407470279230177539421061975684465994591341944...
MATHEMATICA
$MaxExtraPrecision = Infinity;
z = 600; x[0] = E; 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}] (* f(e), f(f(e)), ... *)
RealDigits[x[1]] (* f(e), A229594 *)
Numerator[c[1]] (* A229595 *)
Denominator[c[1]] (* A229596 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 26 2013
STATUS
approved