A228492 Decimal expansion of continued fraction transform of Pi.

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

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: A340461 A060481 A335228 * A340875 A329038 A272491

Adjacent sequences: A228489 A228490 A228491 * A228493 A228494 A228495

Keyword

nonn,cons

Author

Clark Kimberling, Oct 01 2013

Status

approved