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%I #4 Oct 07 2013 04:59:28
%S 2,4,2,5,6,0,1,4,4,0,2,7,7,1,4,5,3,9,9,9,3,8,3,2,6,2,1,7,7,0,1,2,6,2,
%T 6,0,6,4,8,4,7,4,6,5,3,2,9,5,1,1,6,9,4,3,7,7,9,2,3,3,2,9,0,1,6,3,9,2,
%U 2,6,9,9,2,1,5,8,6,1,8,6,5,8,9,7,0,3
%N Decimal expansion of self-generating continued fraction with first term 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(2,0) = 2;
%e c(x,1) = [x, x], so that c(2,1) = 5/2;
%e c(x,2) = [x, x, [x, x]], so that c(2,2) = 29/12;
%e c(x,3) = [x, x, [x, x], [x, x, [x, x]]], so that c(2,3) = 961/396 = 2.4267...;
%e c(2,4)= 45667561/45667561/18267036 = 2.42545...;
%e c(2,5) = 2.454562...;
%e f(2) = 2.4256014402771453999383262177012626064847465329511694...
%t $MaxExtraPrecision = Infinity; z = 300; c[x_, 0] := x; c[x_, n_] := c[x, n] = FromContinuedFraction[Table[c[x, k], {k, 0, n - 1}]]; x = N[2, 300]; t1 = Table[c[x, k], {k, 0, z}]; u = N[c[x, z], 120] (* A229920 *)
%t RealDigits[u]
%Y Cf. A064845, A064846, A229779.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Oct 03 2013
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