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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
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.
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
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 01 2013
STATUS
approved