A229779 Decimal expansion of Quet's constant, the self-generating continued fraction with first term 1.

1, 7, 1, 1, 8, 6, 9, 1, 8, 6, 8, 9, 3, 8, 7, 1, 8, 5, 6, 6, 8, 0, 6, 9, 2, 2, 3, 8, 5, 7, 2, 5, 6, 8, 4, 6, 8, 9, 4, 3, 4, 6, 9, 7, 3, 2, 8, 5, 4, 1, 5, 9, 1, 6, 7, 2, 5, 3, 9, 2, 1, 4, 5, 4, 8, 9, 6, 0, 9, 6, 9, 5, 2, 0, 2, 6, 0, 0, 6, 9, 0, 9, 2, 1, 6, 8

Offset

1,2

Comments

This constant is defined by Leroy Quet at A064845. A generalization follows. 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. In Program 2 in the Mathematica section, f(x) is shown asymptotic to y = x, with local minimum 1.6247... at x = 0.69545...

c(n,x) is a rational function p(n,x)/q(n,x) in which p(n,x) has degree 2^n and q(n,x) has degree 2^n-1; q(n,x) divides q(n+1,x).

Links

Table of n, a(n) for n=1..86.

Example

c(x,0) = x, so that c(1,0) = 1;
c(x,1) = [x, x], so that c(1,1) = 2;
c(x,2) = [x, x, [x, x]], so that c(1,2) = 5/3 = 1.666...;
c(x,3) = [x, x, [x, x], [x, x, [x, x]]], so that c(1,3) = 31/18 = 1.7222...;
c(1,4)= 1231/720 = 1.70972...;
c(1,5) = 4667676800641/2726809311600 = 1.712326...;
f(1) = 1.711869186893871856680692238572...

Mathematica

$MaxExtraPrecision = Infinity;
z = 200; c[x_, 0] := x; c[x_, n_] := c[x, n] = FromContinuedFraction[Table[c[x, k], {k, 0, n - 1}]]; x = N[1, 500]; t1 = Table[c[x, k], {k, 0, z}]; u = N[c[x, z], 120]
RealDigits[u]  (* A229779 *)
(* Program 2:  graph of f(x) *)
c[x_, 0] := x; c[x_, n_] := c[x, n] = FromContinuedFraction[Table[c[x, k], {k, 0, n - 1}]]; Plot[{x, c[x, 20]}, {x, -3, 3}]

Crossrefs

Cf. A053978, A064845, A064846, A229920.

Sequence in context: A367764 A174095 A305607 * A050179 A358450 A183352

Adjacent sequences: A229776 A229777 A229778 * A229780 A229781 A229782

Keyword

nonn,cons

Author

Clark Kimberling, Oct 03 2013

Status

approved