A286984 Decimal expansion of (2 + sqrt(5) + sqrt(15 - 6*sqrt(5)))/2.

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

Offset: 1

Author

Felix Fröhlich, May 17 2017

Keywords

• nonn
• cons

Comments

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))), ....".

A quartic integer with minimal polynomial x^4 - 4x^3 - 4x^2 + 31x - 29. - Charles R Greathouse IV, May 17 2017

Examples

			2.74723827493230433305746518613420282675...
		

References

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

Links

• B. C. Berndt, Y. S. Choi, and S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q722, JIMS VII).
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A239349, A318709.

Programs

Mathematica

RealDigits[(2 + Sqrt[5] + Sqrt[15-6*Sqrt[5]])/2, 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)

PARI

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

PARI

solve(x=2,3,x-sqrt(5+sqrt(5+sqrt(5-sqrt(5 + x))))) \\ Hugo Pfoertner, Sep 02 2018