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Denominators of continued fraction transform of Pi.
4

%I #9 Oct 07 2013 03:45:17

%S 1,3,7,24,79,182,261,965,179751,360467,900685,19274852,20175537,

%T 39450389,59625926,99076315,158702241,1844800966,85019546677,

%U 171883894320,1803858489877,1975742384197,5755343258271,42263145192094,90281633642459,403389679761930

%N Denominators of continued fraction transform of Pi.

%C The function f defined at A229350 is 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), ... ].

%e The first 5 convergents to f(Pi) are 3/1, 10/3, 23/7, 79/24, 260/79.

%t $MaxExtraPrecision = Infinity;

%t 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)), ... *)

%t t1 = RealDigits[x[1]] (* f(Pi), A228493 *)

%t t2 = Numerator[c[1]] (* A228992 *)

%t t3 = Denominator[c[1]] (* A228993 *)

%Y Cf. A228492, A228493, A228993.

%K nonn,frac

%O 1,2

%A _Clark Kimberling_, Oct 01 2013