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.
Decimal expansion of the real root of x^3 + 2*x - 1. - Wolfdieter Lang, Aug 18 2022
LINKS
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
Equals ((1/2)*(1 + sqrt(3*59)/9))^(1/3) - ((1/2)*(1 - sqrt(3*59)/9))^(1/3)*(1 - sqrt(3)*i)/2, with i = sqrt(-1). - Wolfdieter Lang, Aug 19 2022
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)
(PARI) polrootsreal(x^3+2*x-1)[1] \\ Charles R Greathouse IV, Oct 27 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Stanislav Sykora, May 08 2016
STATUS
approved