

A272874


Decimal expansion of the infinite nested radical sqrt(1+sqrt(1+sqrt(1+sqrt(1+ ... ))).


3



4, 5, 3, 3, 9, 7, 6, 5, 1, 5, 1, 6, 4, 0, 3, 7, 6, 7, 6, 4, 4, 7, 4, 6, 5, 3, 9, 0, 0, 0, 1, 9, 2, 1, 8, 8, 8, 6, 6, 8, 8, 4, 4, 2, 4, 9, 6, 5, 0, 7, 7, 6, 5, 9, 8, 8, 1, 6, 6, 3, 2, 8, 5, 4, 3, 2, 3, 3, 3, 2, 3, 0, 4, 2, 1, 1, 6, 8, 6, 0, 5, 6, 6, 7, 8, 7, 2, 5, 1, 4, 8, 4, 9, 6, 4, 0, 5, 9, 9, 7, 6, 3, 1, 5, 3
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OFFSET

0,1


COMMENTS

The radical is intended as follows: let M(z) = sqrt(1+sqrt(1+z)) be an endomorphism on C, with sqrt restricted to its main branch. It has two invariant points which both happen to be real: this value z = a, and z = 0. Moreover, 'a' is an attractor of M(z) which, when iterated, converges to it from any starting complex value except z = 0. Consequently, the nested radical, when truncated after n terms, either stays identically 0 when n is even, or converges to 'a' when n is odd. According to the definition, 'a' is a solution of z = M(z), and therefore a root of the equation z^3 + 2z  1=0.
A closely related case with similar characteristics is the infinite nested radical sqrt(1+sqrt(1+sqrt(1+sqrt(1+ ... ))) which leads to the mapping F(z) = sqrt(1+sqrt(1+z)) instead of M(z), and the value of its respective attractor is A137421.


LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..2000


FORMULA

Satisfies x = sqrt(1+sqrt(1+x)).
Equals 1/6*(108+12*sqrt(177))^(1/3)4/(108+12*sqrt(177))^(1/3).  Alois P. Heinz, May 09 2016


EXAMPLE

0.45339765151640376764474653900019218886688442496507765988166328543...


MATHEMATICA

RealDigits[N[x/.Solve[x == Sqrt[Sqrt[x+1]1], x][[2]], 100]][[1]] (* Giovanni Resta, May 10 2016 *)


PROG

(PARI) real(polroots(Pol([1, 0, 2, 1]))[1])
(PARI) \\ Iterative version; using realprecision of 2100 digits:
M(z)=sqrt(1+sqrt(1+z)); x=1; \\ Starting with a real x>0, all terms are actually real.
\\ Over 6000 iterations were needed to make stable 2000 digits:
for(n=1, 6500, x=M(x)); real(x)


CROSSREFS

Cf. A137421.
Sequence in context: A200351 A138753 A179410 * A008962 A175725 A212711
Adjacent sequences: A272871 A272872 A272873 * A272875 A272876 A272877


KEYWORD

nonn,cons


AUTHOR

Stanislav Sykora, May 08 2016


STATUS

approved



