A228993 Decimal expansion of L(Pi), the limit of iterations of continued fraction transforms of Pi.

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

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: A364832 A349890 A130328 * A083569 A071574 A341336

Adjacent sequences: A228990 A228991 A228992 * A228994 A228995 A228996

Keyword

nonn,cons

Author

Clark Kimberling, Oct 01 2013

Status

approved