A296448 Decimal expansion of the second Ramanujan trigonometric constant r_2.

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

Offset: 0

Author

Vladimir Shevelev, Dec 13 2017

Keywords

• cons
• nonn

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].

Examples

			0.4934146259187856644256727533936734264337374783993750186366641795494767587...
		

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, and L. C. Zhang, Radicals and units in Ramanujan's work, Acta Arith., 87 (1988), 145-158.
• B. C. Berndt, Y. S. Choi, and 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 and S. Bhargava, Ramanujan - for Lowbrows, Amer. Math. Monthly, 100, no. 7, 1993, 644-656.
• Vladimir Shevelev, Three Ramanujan's formulas, Kvant 6 (1988), 52-55 in Russian. English translation: Kvant Selecta 14 (1999), 139-144.
• Vladimir Shevelev, On Ramanujan cubic polynomials, arXiv:0711.3420 [math.AC], 2007; South East Asian J. Math. & Math. Sci. 8 (2009), 113-122.

Crossrefs

Cf. A295872.

Programs

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 *)

PARI

((3*9^(1/3) - 6)/2)^(1/3) \\ Michel Marcus, Dec 13 2017

Formula

r_2 = (3/2 (3^(2/3) -2))^(1/3)

Extensions

More terms from Michel Marcus, Dec 13 2017