A258067 - OEIS

%I #9 May 18 2015 18:47:20

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

%T 7,5,8,9,4,8,5,8,4,3,8,3,3,3,4,5,3,0,2,8,8,7,0,0,0,3,7,3,5,8,9,9,5,6,

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

%N Constant x (second of 2) that satisfies: x = 1 + Sum_{n>=1} frac( x^(n/2) ) / 2^n.

%C In order for a positive x to satisfy: x = 1 + Sum_{n>=1} {x^(n/2)}/2^n, x must be found in the open interval (2^(2/7), 2^(1/3)).

%C When x <= 2^(2/7), then 1 + Sum_{n>=1} {x^(n/2)}/2^n < x ;

%C when x >= 2^(1/3), then 1 + Sum_{n>=1} {x^(n/2)}/2^n > x.

%F Also, x = 1 + Sum_{n>=1} {sqrt(x^n)} / 2^n, where {z} denotes the fractional part of z.

%e x = 1.2517833968605401836370872270577747589485843833345\

%e 30288700037358995625679011795452979128839028921192\

%e 46847369883507572337...

%e The constant is found in the interval (2^(2/7), 2^(1/3)) where

%e 2^(2/7) = 1.219013654204475..., 2^(1/3) = 1.259921049894873...

%o (PARI) x=1.25178; for(i=1,1301,x = (5*x - 1 - sum(n=1,400,frac(x^(n/2))/2^n))/4);x

%Y Cf. A258066.

%K nonn,cons

%O 1,2

%A _Paul D. Hanna_, May 18 2015