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A228993 Decimal expansion of L(Pi), the limit of iterations of continued fraction transforms of Pi. 4
3, 2, 7, 6, 5, 0, 3, 3, 8, 5, 0, 1, 4, 4, 2, 4, 4, 6, 3, 1, 3, 8, 6, 9, 7, 2, 3, 5, 0, 0, 1, 9, 1, 0, 2, 1, 8, 3, 6, 4, 2, 5, 5, 3, 8, 4, 1, 6, 8, 0, 6, 5, 4, 0, 9, 1, 7, 4, 2, 2, 2, 0, 8, 4, 8, 0, 1, 7, 5, 5, 0, 4, 9, 9, 5, 1, 6, 2, 9, 0, 2, 8, 9, 5, 4, 5 (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(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.

Conjecture: if x is an irrational number between 3 and 4, then L(x) = L(Pi).

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

L(Pi) = 3.2765033850144244631386972350019102183642553841680654...

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]]   (* f(Pi), A228493 *)

t2 = Numerator[c[1]]    (* A228992 *)

t3 = Denominator[c[1]]  (* A228993 *)

CROSSREFS

Cf. A228492, A229597 (L(e)).

Sequence in context: A201566 A072764 A130328 * A083569 A071574 A276344

Adjacent sequences:  A228990 A228991 A228992 * A228994 A228995 A228996

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 01 2013

STATUS

approved

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Last modified January 19 20:01 EST 2019. Contains 319309 sequences. (Running on oeis4.)