OFFSET
0,1
COMMENTS
According to the famous Ramanujan identity, the constant r_2 has a representation: r_2 = Sum_{i = 1..3} (cos(2^i*Pi/9))^(1/3) (see formula). This identity was submitted in 1914 by Ramanujan as a problem (cf. [Berndt, Y. S. Choi, S. Y. Kang]). For proof, see first [V. Shevelev].
REFERENCES
B. Bajorska-Harapinska, M. Pleszczynski, D. Slota and R. Witula, A few properties of Ramanujan cubic polynomials and Ramanujan cubic polynomials of the second kind, in book: Selected Problems on Experimental Mathematics, Gliwice 2017, pp. 181-200.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
LINKS
B. C. Berndt, H. H. Chan, L. C. Zhang, Radicals and units in Ramanujan's work, Acta Arith., 87 (1988), 145-158.
B. C. Berndt, Y. S. Choi, 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 Q524, JIMS VI, 1914).
B. C. Berndt, S. Bhargava, Ramanujan - for Lowbrows, Amer. Math. Monthly, 100, no. 7, 1993, 644-656.
V. Shevelev, Three Ramanujan's formulas, Kvant 6 (1988), 52-55 in Russian. English translation: Kvant Selecta 14 (1999), 139-144.
V. Shevelev, On Ramanujan cubic polynomials, arXiv:0711.3420 [math.AC], 2007; South East Asian J. Math. & Math. Sci. 8 (2009), 113-122.
FORMULA
r_2 = (3/2 (3^(2/3) -2))^(1/3)
EXAMPLE
0.4934146259187856644256727533936734264337374783993750186366641795494767587...
MAPLE
use RealDomain in solve(8*x^9 + 72*x^6 + 216*x^3 - 27 = 0) end use:
evalf(%, 85); # Peter Luschny, Dec 13 2017
MATHEMATICA
RealDigits[(3/2 (-2 + 3^(2/3)))^(1/3), 10, 111][[1]] (* Robert G. Wilson v, Dec 13 2017 *)
PROG
(PARI) ((3*9^(1/3) - 6)/2)^(1/3) \\ Michel Marcus, Dec 13 2017
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Vladimir Shevelev, Dec 13 2017
EXTENSIONS
More terms from Michel Marcus, Dec 13 2017
STATUS
approved