Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
editing
approved
<a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
(PARI) polrootsreal(x^3+2*x-1)[1] \\ Charles R Greathouse IV, Oct 27 2023
approved
editing
reviewed
approved
proposed
reviewed
editing
proposed
Decimal expansion of the infinite nested radical sqrt(-1 + sqrt(1 + sqrt(-1 + sqrt(1 + ... ))).
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.
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
proposed
editing
editing
proposed
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
approved
editing
editing
approved
Satisfies x = sqrt(-1+sqrt(1+x)). For x real and positive.