A229779 - OEIS

%I #36 Feb 11 2014 02:23:33

%S 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,

%T 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,

%U 9,6,9,5,2,0,2,6,0,0,6,9,0,9,2,1,6,8

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

%C 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 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).

%e c(x,0) = x, so that c(1,0) = 1;

%e c(x,1) = [x, x], so that c(1,1) = 2;

%e c(x,2) = [x, x, [x, x]], so that c(1,2) = 5/3 = 1.666...;

%e c(x,3) = [x, x, [x, x], [x, x, [x, x]]], so that c(1,3) = 31/18 = 1.7222...;

%e c(1,4)= 1231/720 = 1.70972...;

%e c(1,5) = 4667676800641/2726809311600 = 1.712326...;

%e f(1) = 1.711869186893871856680692238572...

%t $MaxExtraPrecision = Infinity;

%t 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]

%t RealDigits[u]  (* A229779 *)

%t (* Program 2:  graph of f(x) *)

%t 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}]

%Y Cf. A053978, A064845, A064846, A229920.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Oct 03 2013