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).

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

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 03 2013

STATUS

approved