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A229779 Decimal expansion of Quet's constant, the self-generating continued fraction with first term 1. 7
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 (list; constant; graph; refs; listen; history; text; internal format)
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: A336459 A174095 A305607 * A050179 A183352 A217510

Adjacent sequences:  A229776 A229777 A229778 * A229780 A229781 A229782

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 03 2013

STATUS

approved

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Last modified September 27 13:39 EDT 2020. Contains 337380 sequences. (Running on oeis4.)