A258066 Constant x (first of 2) that satisfies: x = 1 + Sum_{n>=1} frac( x^(n/2) ) / 2^n.

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.

Links

Table of n, a(n) for n=1..120.

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

Adjacent sequences: A258063 A258064 A258065 * A258067 A258068 A258069

Keyword

nonn,cons

Author

Paul D. Hanna, May 18 2015

Status

approved