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%I #5 Sep 30 2013 12:16:50
%S 2,2,9,9,9,1,4,5,9,3,8,5,9,4,6,2,1,9,7,8,6,7,5,2,0,4,6,5,2,7,0,0,2,7,
%T 6,8,1,5,2,3,3,1,3,6,5,2,8,0,4,8,2,5,0,7,1,7,1,7,9,5,2,1,4,2,9,8,1,6,
%U 4,4,7,5,0,7,4,7,0,5,5,4
%N Decimal expansion 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), ... ].
%C Thus, f(e) = 2.9991459..., f(f(e)) = 2.3690966..., f(f(f(e))) = 2.3483570...; 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(e) = 2.34840747027923017..., as in A229597.
%e f(e) = 2.29991459385946219786752046527002768152331365280482...
%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. A229595, A229596, A229597.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Sep 26 2013
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