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Numerators of continued fraction transform of Pi.
4

%I #5 Oct 02 2013 15:09:32

%S 3,10,23,79,260,599,859,3176,591595,1186366,2964327,63437233,66401560,

%T 129838793,196240353,326079146,522319499,6071593635,279815626709,

%U 565702847053,5936844097239,6502546944292,18941937985823,139096112845053,297134163675929

%N Numerators 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. A229594, A228493, A228992, A228993.

%K nonn,frac

%O 1,1

%A _Clark Kimberling_, Oct 01 2013