A317969 Decimal expansion of (2^(1/3)-1)^(1/3).

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

Offset

0,1

Comments

(2^(1/3)-1)^(1/3) = (1/9)^(1/3) - (2/9)^(1/3) + (4/9)^(1/3) is a famous and remarkable identity of Ramanujan's.

Ramanujan's question 1076 (ii), see Berndt and Rankin in References: Show that (4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8) = (4/9)^(1/3)-(2/9)^(1/3)+(1/9)^(1/3). - Hugo Pfoertner, Aug 28 2018

References

B. C. Berndt and R. A. Rankin, Ramanujan: Essays and Surveys, American Mathematical Society, 2001, ISBN 0-8218-2624-7, page 222 (JIMS 11, page 199).

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.1.2, p. 4.

S. Ramanujan, Coll. Papers, Chelsea, 1962, page 331, Question 682; page 334 Question 1076.

Links

Table of n, a(n) for n=0..84.

Susan Landau, Simplification of nested radicals, SIAM Journal on Computing 21.1 (1992): 85-110. See page 85. [Do not confuse this paper with the short FOCS conference paper with the same title, which is only a few pages long.]

S. Ramanujan, Question 682, Journal of the Indian Mathematical Society, VII, p. 160.

S. Ramanujan, Question 1076, Journal of the Indian Mathematical Society, XI, p. 199.

Vincent Thill, Radicaux et Ramanujan, April 2021, see k.

Formula

From Michel Marcus, Jan 08 2022: (Start)

Equals (A002580-1)^(1/3).

k^(3*n) = x(n) + A002580*y(n) + A005480*z(n) where k is this constant z(n) = A108369(n-1), y(n) = z(n)+z(n+1), x(n) = y(n)+y(n+1); A002580 and A005480 are the cube root of 2 and 4. (End)

Minimal polynomial: 1 - 3*x^3 - 3*x^6 - x^9. - Stefano Spezia, Oct 15 2024

Example

0.638185820860644153015503659444067701265157543977997683421...

Maple

evalf((4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8)); # Muniru A Asiru, Aug 28 2018

Mathematica

RealDigits[N[Power[Power[2, (3)^-1] - 1, (3)^-1], 100]] (* Peter Cullen Burbery, Apr 09 2022 *)

Prog

(PARI) (4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8) /* Hugo Pfoertner Aug 28 2018 */
(PARI) sqrtn(1/9, 3) - sqrtn(2/9, 3) + sqrtn(4/9, 3) \\ Michel Marcus, Jan 07 2022

Crossrefs

Cf. A318526.

Cf. A002580, A005480, A108369.

Sequence in context: A182566 A115287 A061533 * A281682 A157294 A021098

Adjacent sequences: A317966 A317967 A317968 * A317970 A317971 A317972

Keyword

nonn,cons

Author

N. J. A. Sloane, Aug 27 2018

Status

approved