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%I #4 Oct 07 2013 21:25:34
%S 1,6,9,7,3,0,4,4,7,0,0,7,1,2,8,2,6,9,4,3,1,2,5,1,0,9,4,1,9,4,9,5,6,5,
%T 8,4,1,7,0,1,3,2,0,8,6,3,5,5,4,3,2,9,9,2,7,0,0,9,6,0,2,8,3,0,8,9,2,5,
%U 3,3,9,4,2,5,2,2,6,1,1,6,7,9,7,0,8,9,4,1,0,8,9,0,4,1,4,4,4,8,9,3,9,3,2,6,7,4,5,4,0,8,0,6,9,2,8,4,8,4,1,1
%N Decimal expansion of self-generating continued fraction with first term 1/2.
%C For x > 0, define c(x,0) = x and c(x,n) = [c(x,0), ..., c(x,n-1)]. We call f(x) the self-generating continued fraction with first term x. See A229779.
%e c(x,0) = x, so that c(1/2,0) = 1/2;
%e c(x,1) = [x, x], so that c(1/2,1) = 5/2;
%e c(x,2) = [x, x, [x, x]], so that c(1/2,2) = 29/18 = 1.6111...;
%e c(x,3) = [x, x, [x, x], [x, x, [x, x]]], so that c(1/2,3) = 1021/594 = 1.718...;
%e c(1/2,4) = 1352509/798930 = 1.6929...
%e f(1/2) = 1.697304470071282694312510941949565841701320863554...
%t $MaxExtraPrecision = Infinity; z = 300; c[x_, 0] := x; c[x_, n_] :=
%t c[x, n] = FromContinuedFraction[Table[c[x, k], {k, 0, n - 1}]]; x = N[1/2, 300]; t1 = Table[c[x, k], {k, 0, z}]; u = N[c[x, z], 120] (* A229922 *)
%t RealDigits[u]
%Y Cf. A064845, A064846, A229779.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Oct 03 2013
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