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

0,1

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].

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, 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

Table of n, a(n) for n=0..72.

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, 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_1 = ((5 - 3*7^(1/3))/2)^(1/3).

Example

r_1 =-0.7175150796499399351209505591779861121084576011552505721833028300279814650...

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 *)

Prog

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

Crossrefs

Sequence in context: A198580 A160798 A033953 * A351835 A010772 A199732

Adjacent sequences: A295869 A295870 A295871 * A295873 A295874 A295875

Keyword

cons,nonn

Author

Vladimir Shevelev, Dec 09 2017

Extensions

More terms from Michel Marcus, Dec 09 2017

Status

approved