OFFSET
1,2
COMMENTS
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)).
When x <= 2^(2/7), then 1 + Sum_{n>=1} {x^(n/2)}/2^n < x ;
when x >= 2^(1/3), then 1 + Sum_{n>=1} {x^(n/2)}/2^n > x.
FORMULA
Also, x = 1 + Sum_{n>=1} {sqrt(x^n)} / 2^n, where {z} denotes the fractional part of z.
EXAMPLE
x = 1.2451046148931201562402697069856521205921473073362\
91950805115511837651892130077964694889192025530949\
56446692373125951761...
The constant is found in the interval (2^(2/7), 2^(1/3)) where
2^(2/7) = 1.219013654204475..., 2^(1/3) = 1.259921049894873...
PROG
(PARI) x=1.2451; for(i=1, 1301, x = (5*x - 1 - sum(n=1, 400, frac(x^(n/2))/2^n))/4); x
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Paul D. Hanna, May 18 2015
STATUS
approved