A229594 Decimal expansion of continued fraction transform of e; see Comments.

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

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) = 2.34840747027923017..., as in A229597.

Links

Table of n, a(n) for n=1..80.

Example

f(e) = 2.29991459385946219786752046527002768152331365280482...

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. A229595, A229596, A229597.

Sequence in context: A104681 A056856 A133920 * A059199 A010765 A193477

Adjacent sequences: A229591 A229592 A229593 * A229595 A229596 A229597

Keyword

nonn,cons

Author

Clark Kimberling, Sep 26 2013

Status

approved