A229921 Decimal expansion of self-generating continued fraction with first term 1/2.

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, 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, 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

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..120.

Example

c(x,0) = x, so that c(1/2,0) = 1/2;
c(x,1) = [x, x], so that c(1/2,1) = 5/2;
c(x,2) = [x, x, [x, x]], so that c(1/2,2) = 29/18 = 1.6111...;
c(x,3) = [x, x, [x, x], [x, x, [x, x]]], so that c(1/2,3) = 1021/594 =  1.718...;
c(1/2,4) = 1352509/798930 = 1.6929...
f(1/2) = 1.697304470071282694312510941949565841701320863554...

Mathematica

$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[1/2, 300]; t1 = Table[c[x, k], {k, 0, z}]; u = N[c[x, z], 120] (* A229922 *)
RealDigits[u]

Crossrefs

Cf. A064845, A064846, A229779.

Sequence in context: A011480 A283743 A339764 * A145429 A194789 A273082

Adjacent sequences: A229918 A229919 A229920 * A229922 A229923 A229924

Keyword

nonn,cons

Author

Clark Kimberling, Oct 03 2013

Status

approved