%I #24 Oct 27 2023 10:46:54

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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.

%C Decimal expansion of the real root of x^3 + 2*x - 1. - _Wolfdieter Lang_, Aug 18 2022

%H Stanislav Sykora, <a href="/A272874/b272874.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>

%F Satisfies x = sqrt(-1 + sqrt(1+x)).

%F Equals (1/6)*(108 + 12*sqrt(177))^(1/3) - 4/(108 + 12*sqrt(177))^(1/3). - _Alois P. Heinz_, May 09 2016

%F 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

%e 0.45339765151640376764474653900019218886688442496507765988166328543...

%t RealDigits[N[x/.Solve[x == Sqrt[Sqrt[x+1]-1], x][[2]], 100]][[1]] (* _Giovanni Resta_, May 10 2016 *)

%o (PARI) real(polroots(Pol([1,0,2,-1]))[1])

%o (PARI) \\ Iterative version; using realprecision of 2100 digits:

%o M(z)=sqrt(-1+sqrt(1+z)); x=1; \\ Starting with a real x>0, all terms are actually real.

%o \\ Over 6000 iterations were needed to make stable 2000 digits:

%o for(n=1,6500,x=M(x));real(x)

%o (PARI) polrootsreal(x^3+2*x-1)[1] \\ _Charles R Greathouse IV_, Oct 27 2023

%Y Cf. A137421.

%K nonn,cons

%O 0,1

%A _Stanislav Sykora_, May 08 2016