

A229597


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


4



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(n1,x)). Then L(e) = 2.34840747..., .
Conjecture: if x is an irrational number between 2 and 3, then L(x) = L(e).


LINKS



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 *)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



