A286984 - OEIS

%I #24 May 01 2024 17:02:30

%S 2,7,4,7,2,3,8,2,7,4,9,3,2,3,0,4,3,3,3,0,5,7,4,6,5,1,8,6,1,3,4,2,0,2,

%T 8,2,6,7,5,8,1,6,3,8,7,8,7,7,6,1,6,7,9,8,7,7,8,3,8,0,4,3,7,3,8,5,6,2,

%U 2,4,3,6,4,8,5,3,8,3,0,1,5,0,3,4,3,1,5

%N Decimal expansion of (2 + sqrt(5) + sqrt(15 - 6*sqrt(5)))/2.

%C See Question 722 on page 219 of Berndt and Rankin, 2001. This says, in part: "Solve completely x^2 = a + y, y^2 = a + z, z^2 = a + u, u^2 = a + x and deduce that, if x = sqrt(5 + sqrt(5 + sqrt(5 - sqrt(5 + x)))), then x = 1/2(2 + sqrt(5) + sqrt(15 - 6*sqrt(5))), ....".

%C A quartic integer with minimal polynomial x^4 - 4x^3 - 4x^2 + 31x - 29. - _Charles R Greathouse IV_, May 17 2017

%D B. C. Berndt and R. A. Rankin, Ramanujan: Essays and Surveys, American Mathematical Society, 2001, ISBN 0-8218-2624-7.

%H B. C. Berndt, Y. S. Choi, and S. Y. Kang, <a href="https://faculty.math.illinois.edu/~berndt/jims.ps">The problems submitted by Ramanujan to the Journal of Indian Math. Soc.</a>, in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q722, JIMS VII).

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>

%e 2.74723827493230433305746518613420282675...

%t RealDigits[(2 + Sqrt[5] + Sqrt[15-6*Sqrt[5]])/2, 10, 120][[1]] (* _Amiram Eldar_, Jun 27 2023 *)

%o (PARI) default(realprecision, 90); (2+sqrt(5)+sqrt(15-6*sqrt(5)))/2

%o (PARI) solve(x=2,3,x-sqrt(5+sqrt(5+sqrt(5-sqrt(5 + x))))) \\ _Hugo Pfoertner_, Sep 02 2018

%Y Cf. A239349, A318709.

%K nonn,cons

%O 1,1

%A _Felix Fröhlich_, May 17 2017