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A228492 Decimal expansion of continued fraction transform of Pi. 4
3, 2, 9, 1, 1, 9, 1, 7, 0, 4, 0, 9, 1, 9, 4, 1, 5, 9, 1, 0, 5, 2, 0, 2, 8, 2, 3, 8, 7, 5, 1, 1, 0, 4, 5, 4, 7, 0, 0, 4, 5, 5, 2, 7, 5, 6, 2, 0, 3, 9, 1, 5, 8, 6, 8, 9, 9, 0, 4, 5, 1, 1, 6, 8, 2, 5, 5, 7, 6, 0, 8, 8, 1, 3, 5, 9, 7, 6, 9, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The function f defined at A229350 is the continued fraction transform; 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(pi) = 3.291191..., f(f(pi)) = 3.276718..., f(f(f(pi))) =3.276503 ...; 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(pi) = 3.276503 ..., as in A228993.

LINKS

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

EXAMPLE

f(pi) = 3.29119170409194159105202823875110454700455275620391586...

MATHEMATICA

$MaxExtraPrecision = Infinity;

z = 600; x[0] = Pi; 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}] (* A228492, f(pi), f(f(pi)), ... *)

t1 = RealDigits[x[1]]   (* A228493 *)

t2 = Numerator[c[1]]    (* A228992 *)

t3 = Denominator[c[1]]  (* A228993 *)

CROSSREFS

Cf. A228493, A228992, A228993.

Sequence in context: A211878 A060481 A335228 * A329038 A272491 A010271

Adjacent sequences:  A228489 A228490 A228491 * A228493 A228494 A228495

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 01 2013

STATUS

approved

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Last modified September 24 01:21 EDT 2020. Contains 337315 sequences. (Running on oeis4.)