

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
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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. 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.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.


LINKS

Table of n, a(n) for n=0..104.
B. C. Berndt, H. H. Chan, L. C. Zhang, Radicals and units in Ramanujan's work, Acta Arith., 87 (1988), 145158.
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).
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_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



