

A229779


Decimal expansion of Quet's constant, the selfgenerating 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
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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,n1)]. We call f(x) the selfgenerating 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^n1; q(n,x) divides q(n+1,x).


LINKS



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]
(* 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}]


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STATUS

approved



