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 A296448 Decimal expansion of the second Ramanujan trigonometric constant r_2. 0
 4, 9, 3, 4, 1, 4, 6, 2, 5, 9, 1, 8, 7, 8, 5, 6, 6, 4, 4, 2, 5, 6, 7, 2, 7, 5, 3, 3, 9, 3, 6, 7, 3, 4, 2, 6, 4, 3, 3, 7, 3, 7, 4, 7, 8, 3, 9, 9, 3, 7, 5, 0, 1, 8, 6, 3, 6, 6, 6, 4, 1, 7, 9, 5, 4, 9, 4, 7, 6, 7, 5, 8, 7, 8, 7, 8, 5, 9, 1, 8, 0, 5, 7, 4, 3, 2, 5, 1, 6, 9, 4, 1, 2, 9, 4, 5, 9, 7, 2, 4, 2, 8, 4, 0, 9 (list; constant; graph; refs; listen; history; text; internal format)
 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 Cf. A295872. Sequence in context: A021957 A096301 A196819 * A217316 A159628 A102753 Adjacent sequences:  A296445 A296446 A296447 * A296449 A296450 A296451 KEYWORD cons,nonn AUTHOR Vladimir Shevelev, Dec 13 2017 EXTENSIONS More terms from Michel Marcus, Dec 13 2017 STATUS approved

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Last modified April 8 14:52 EDT 2020. Contains 333314 sequences. (Running on oeis4.)