A295872 Decimal expansion of the first Ramanujan trigonometric constant (negated).

7, 1, 7, 5, 1, 5, 0, 7, 9, 6, 4, 9, 9, 3, 9, 9, 3, 5, 1, 2, 0, 9, 5, 0, 5, 5, 9, 1, 7, 7, 9, 8, 6, 1, 1, 2, 1, 0, 8, 4, 5, 7, 6, 0, 1, 1, 5, 5, 2, 5, 0, 5, 7, 2, 1, 8, 3, 3, 0, 2, 8, 3, 0, 0, 2, 7, 9, 8, 1, 4, 6, 5, 0

Offset: 0

Author

Vladimir Shevelev, Dec 09 2017

Keywords

• cons
• nonn

Comments

According to the famous Ramanujan identity, the constant r_1 has a representation: r_1 = Sum_{i = 1..3} (cos(2^i*Pi/7))^(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

			r_1 =-0.7175150796499399351209505591779861121084576011552505721833028300279814650...
		

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.
• 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).
• 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 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.

Programs

Maple

use RealDomain in solve(4*x^9 - 30*x^6 + 75*x^3 + 32 = 0) end use:
evalf(%, 79); # Peter Luschny, Dec 13 2017

Mathematica

RealDigits[(-(5 - 3*7^(1/3))/2)^(1/3), 10, 111][[1]] (* Robert G. Wilson v, Dec 13 2017 *)

PARI

-((3*7^(1/3)-5)/2)^(1/3) \\ Michel Marcus, Dec 10 2017

Formula

r_1 = ((5 - 3*7^(1/3))/2)^(1/3).

Extensions

More terms from Michel Marcus, Dec 09 2017