A229597 Decimal expansion of L(e), the limit of iterations of continued fraction transforms of e.

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, 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, 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

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).

Links

Table of n, a(n) for n=1..116.

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

Cf. A229594, A229595, A229596.

Sequence in context: A109617 A181881 A071373 * A366656 A175060 A138773

Adjacent sequences: A229594 A229595 A229596 * A229598 A229599 A229600

Keyword

nonn,cons

Author

Clark Kimberling, Sep 26 2013

Status

approved