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A229595 Numerators of continued fraction transform of e; see Comments. 4

%I #5 Sep 30 2013 12:17:12

%S 2,7,16,23,2684,5391,8075,13466,35007,48473,83480,131953,347386,

%T 7079673,7427059,14506732,21933791,189977060,211910851,613798762,

%U 1439508375,2053307137,5546122649,96337392170,1739619181709,20971767572678,22711386754387,66394541081452

%N Numerators of continued fraction transform of e; see Comments.

%C 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), ... ].

%e The first 5 convergents to f(e) are 2/1, 7/3, 16/7, 23/10, 2684/1167.

%t $MaxExtraPrecision = Infinity;

%t z = 600; x[0] = E; c[0] = Convergents[x[0], z];

%t x[n_] := N[FromContinuedFraction[c[n - 1]], 80];

%t c[n_] := Convergents[x[n]];

%t Table[x[n], {n, 1, 20}] (* f(e), f(f(e)), ... *)

%t RealDigits[x[1]] (* f(e), A229594 *)

%t Numerator[c[1]] (* A229595 *)

%t Denominator[c[1]] (* A229596 *)

%Y Cf. A229594, A229596, A229597.

%K nonn,frac

%O 0,1

%A _Clark Kimberling_, Sep 26 2013

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