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%I #24 Apr 11 2022 11:35:29
%S 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,
%T 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,
%U 5,4,8,5,1,3,9,7,0,7,9,2,5,0,2,7,6
%N Decimal expansion of (2^(1/3)-1)^(1/3).
%C (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.
%C 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
%D 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).
%D S. Ramanujan, Coll. Papers, Chelsea, 1962, page 331, Question 682; page 334 Question 1076.
%H Susan Landau, <a href="https://www.researchgate.net/publication/2629046_Simplification_of_Nested_Radicals">Simplification of nested radicals</a>, 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.]
%H S. Ramanujan, <a href="https://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q682.htm">Question 682</a>, Journal of the Indian Mathematical Society, VII, p. 160.
%H S. Ramanujan, <a href="https://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q1076.htm">Question 1076</a>, Journal of the Indian Mathematical Society, XI, p. 199.
%H Vincent Thill, <a href="http://vincent-thill.fr/2021/04/identite-du-mois-davril-2021/">Radicaux et Ramanujan</a>, April 2021, see k.
%F From _Michel Marcus_, Jan 08 2022: (Start)
%F Equals (A002580-1)^(1/3).
%F 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)
%e 0.638185820860644153015503659444067701265157543977997683421...
%p evalf((4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8)); # _Muniru A Asiru_, Aug 28 2018
%t RealDigits[N[Power[Power[2, (3)^-1] - 1, (3)^-1], 100]] (* _Peter Cullen Burbery_, Apr 09 2022 *)
%o (PARI) (4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8) /* _Hugo Pfoertner_ Aug 28 2018 */
%o (PARI) sqrtn(1/9, 3) - sqrtn(2/9, 3) + sqrtn(4/9, 3) \\ _Michel Marcus_, Jan 07 2022
%Y Cf. A318526.
%Y Cf. A002580, A005480, A108369.
%K nonn,cons
%O 0,1
%A _N. J. A. Sloane_, Aug 27 2018
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