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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

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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) = 2.34840747027923017..., 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 *)