|
|
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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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:
|
|
MATHEMATICA
|
RealDigits[(-(5 - 3*7^(1/3))/2)^(1/3), 10, 111][[1]] (* Robert G. Wilson v, Dec 13 2017 *)
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|