A229597 - OEIS

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

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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 *)