A318709 Decimal expansion of the solution to x = sqrt(5 + sqrt(5 - sqrt(5 - sqrt(5 + x)))).

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

Offset

1,1

Comments

The last part of Ramanujan's question 722 in the Journal of the Indian Mathematical Society (VII, 240) asked "... deduce that, if x = sqrt(5 + sqrt(5 - sqrt(5 - sqrt(5 + x)))), then x = (1/4) * (sqrt(5) - 2 + sqrt(13 - 4 * sqrt(5)) + sqrt(50 + 12 * sqrt(5) - 2 * sqrt(65 - 20 * sqrt(5))))".

Links

Table of n, a(n) for n=1..78.

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).

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 8.

Formula

Minimal polynomial: x^8 + 4*x^7 - 10*x^6 - 54*x^5 + 9*x^4 + 226*x^3 + 125*x^2 - 301*x - 269. - Amiram Eldar, Jun 05 2026

Example

2.6214083830758615056984952806122431277979706147211676796641678...

Mathematica

RealDigits[(Sqrt[5] - 2 + Sqrt[13 - 4*Sqrt[5]] + Sqrt[50 + 12*Sqrt[5] - 2*Sqrt[65 - 20*Sqrt[5]]])/4, 10, 120][[1]] (* Amiram Eldar, Jun 05 2026 *)

Prog

(PARI) solve(x=2, 3, x-sqrt(5+sqrt(5-sqrt(5-sqrt(5+x)))))
(PARI) (1/4)*(sqrt(5)-2+sqrt(13-4*sqrt(5))+sqrt(50+12*sqrt(5)-2*sqrt(65-20*sqrt(5))))
(PARI) polrootsreal(x^8 + 4*x^7 - 10*x^6 - 54*x^5 + 9*x^4 + 226*x^3 + 125*x^2 - 301*x - 269)[8] \\ Charles R Greathouse IV, May 19 2026

Crossrefs

Cf. A286984.

Sequence in context: A342670 A298782 A082516 * A204935 A008905 A136760

Adjacent sequences: A318706 A318707 A318708 * A318710 A318711 A318712

Keyword

nonn,cons

Author

Hugo Pfoertner, Sep 01 2018

Status

approved