login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A296448 Decimal expansion of the second Ramanujan trigonometric constant r_2. 0

%I

%S 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,

%T 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,

%U 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

%N Decimal expansion of the second Ramanujan trigonometric constant r_2.

%C 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].

%D 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.

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

%H B. C. Berndt, H. H. Chan, L. C. Zhang, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa87/aa8725.pdf">Radicals and units in Ramanujan's work</a>, Acta Arith., 87 (1988), 145-158.

%H B. C. Berndt, Y. S. Choi, S. Y. Kang, <a href="https://faculty.math.illinois.edu/~berndt/jims.ps">The problems submitted by Ramanujan to the Journal of Indian Math. Soc.</a>, in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q524, JIMS VI, 1914).

%H B. C. Berndt, S. Bhargava, <a href="https://www.maa.org/sites/default/files/images/upload_library/22/Ford/Berndt-Bhargava644-656.pdf">Ramanujan - for Lowbrows</a>, Amer. Math. Monthly, 100, no. 7, 1993, 644-656.

%H V. Shevelev, <a href="http://kvant.mccme.ru/1988/06/tri_formuly_ramanudzhana.htm">Three Ramanujan's formulas</a>, Kvant 6 (1988), 52-55 in Russian. English translation: Kvant Selecta 14 (1999), 139-144.

%H V. Shevelev, <a href="https://arxiv.org/abs/0711.3420">On Ramanujan cubic polynomials</a>, arXiv:0711.3420 [math.AC], 2007; South East Asian J. Math. & Math. Sci. 8 (2009), 113-122.

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

%e 0.4934146259187856644256727533936734264337374783993750186366641795494767587...

%p use RealDomain in solve(8*x^9 + 72*x^6 + 216*x^3 - 27 = 0) end use:

%p evalf(%, 85); # _Peter Luschny_, Dec 13 2017

%t RealDigits[(3/2 (-2 + 3^(2/3)))^(1/3), 10, 111][[1]] (* _Robert G. Wilson v_, Dec 13 2017 *)

%o (PARI) ((3*9^(1/3) - 6)/2)^(1/3) \\ _Michel Marcus_, Dec 13 2017

%Y Cf. A295872.

%K cons,nonn

%O 0,1

%A _Vladimir Shevelev_, Dec 13 2017

%E More terms from _Michel Marcus_, Dec 13 2017

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 19 17:16 EDT 2019. Contains 324222 sequences. (Running on oeis4.)