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
Examples
0.638185820860644153015503659444067701265157543977997683421...
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.
Programs
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Maple
evalf((4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8)); # Muniru A Asiru, Aug 28 2018
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Mathematica
RealDigits[N[Power[Power[2, (3)^-1] - 1, (3)^-1], 100]] (* Peter Cullen Burbery, Apr 09 2022 *)
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PARI
(4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8) /* Hugo Pfoertner Aug 28 2018 */
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PARI
sqrtn(1/9, 3) - sqrtn(2/9, 3) + sqrtn(4/9, 3) \\ Michel Marcus, Jan 07 2022
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
Comments