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A317969
Decimal expansion of (2^(1/3)-1)^(1/3).
1
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
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
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane, Aug 27 2018
STATUS
approved