A272874 Decimal expansion of the infinite nested radical sqrt(-1 + sqrt(1 + sqrt(-1 + sqrt(1 + ...)))).

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

Offset: 0

Author

Stanislav Sykora, May 08 2016

Keywords

• nonn
• cons

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

Examples

			0.45339765151640376764474653900019218886688442496507765988166328543...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A137421.

Programs

Mathematica

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

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

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