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A258066
Constant x (first of 2) that satisfies: x = 1 + Sum_{n>=1} frac( x^(n/2) ) / 2^n.
1
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, 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, 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
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
Cf. A258067.
Sequence in context: A198505 A173163 A021412 * A036501 A225153 A360108
KEYWORD
nonn,cons
AUTHOR
Paul D. Hanna, May 18 2015
STATUS
approved