A258066 - OEIS

%I #10 May 18 2015 18:47:36

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

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

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

%N Constant x (first 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 = 1.2451046148931201562402697069856521205921473073362\

%e 91950805115511837651892130077964694889192025530949\

%e 56446692373125951761...

%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=1.2451; for(i=1,1301,x = (5*x - 1 - sum(n=1,400,frac(x^(n/2))/2^n))/4);x

%Y Cf. A258067.

%K nonn,cons

%O 1,2

%A _Paul D. Hanna_, May 18 2015