A258067 Constant x (second of 2) that satisfies: x = 1 + Sum_{n>=1} frac( x^(n/2) ) / 2^n.

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, 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, 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

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 = 1.2517833968605401836370872270577747589485843833345\
30288700037358995625679011795452979128839028921192\
46847369883507572337...
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=1.25178; for(i=1, 1301, x = (5*x - 1 - sum(n=1, 400, frac(x^(n/2))/2^n))/4); x

Crossrefs

Cf. A258066.

Sequence in context: A060789 A134570 A246169 * A240241 A019510 A124576

Adjacent sequences: A258064 A258065 A258066 * A258068 A258069 A258070

Keyword

nonn,cons

Author

Paul D. Hanna, May 18 2015

Status

approved