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A229594 Decimal expansion of continued fraction transform of e; see Comments. 5
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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Last modified June 28 03:08 EDT 2017. Contains 288813 sequences.