

A295872


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


1



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
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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. BajorskaHarapinska, 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. 181200.
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), 1556 (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), 145158.
B. C. Berndt, S. Bhargava, Ramanujan  for Lowbrows, Amer. Math. Monthly, 100, no. 7, 1993, 644656.
V. Shevelev, Three Ramanujan's formulas, Kvant 6 (1988), 5255 in Russian. English translation: Kvant Selecta 14 (1999), 139144.
V. Shevelev, On Ramanujan cubic polynomials, arXiv:0711.3420 [math.AC], 2007; South East Asian J. Math. & Math. Sci. 8 (2009), 113122.


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 * A010772 A199732 A293238
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



