

A318732


Decimal expansion of the nontrivial real solution to x^6  x^3 + x^2 + 2*x  1 = 0.


1



4, 4, 1, 8, 0, 4, 2, 6, 3, 2, 7, 0, 7, 6, 5, 3, 2, 1, 5, 6, 7, 1, 1, 9, 4, 3, 9, 3, 9, 6, 8, 8, 9, 0, 0, 5, 1, 4, 9, 3, 7, 4, 9, 4, 0, 9, 0, 9, 2, 4, 7, 5, 4, 1, 7, 7, 7, 6, 6, 0, 4, 8, 2, 9, 9, 7, 4, 3, 9, 2, 9, 2, 3, 4, 3, 4, 1, 7, 5, 6, 0, 9, 3, 3, 7, 6, 6
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OFFSET

0,1


COMMENTS

The first part of Ramanujan's question 699 in the Journal of the Indian Mathematical Society (VII, 199) asked "Show that the roots of the equations x^6  x^3 + x^2 + 2*x  1 = 0, ... can be expressed in terms of radicals."
The polynomial includes a trivial factor, i.e., x^6  x^3 + x^2 + 2*x  1 = (x + 1) * (x^5  x^4 + x^3  2*x^2 + 3*x  1).


REFERENCES

V. M. Galkin, O. R. Kozyrev, On an algebraic problem of Ramanujan, pp. 8994 in Number Theoretic And Algebraic Methods In Computer Science  Proceedings Of The International Conference, Moscow 1993, Ed. Horst G. Zimmer, World Scientific, 31 Aug 1995


LINKS

Table of n, a(n) for n=0..86.
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 Q699, JIMS VII).


FORMULA

Equals 2^(1/4) / G(79), where G(n) is Ramanujan's class invariant G(n) = 2^(1/4) * q(n)^(1/24) * Product_{k>=0} (1 + q(n)^(2*k + 1)), with q(n) = exp(Pi * sqrt(n)).


EXAMPLE

0.441804263270765321567119439396889005149374940909247541777660...


PROG

(PARI) p(x)=x^5x^4+x^32*x^2+3*x1; solve(x=0.3, 0.5, p(x))
(PARI) q(x)=exp(Pi*sqrt(x)); G(n)=2^(1/4)*q(n)^(1/24)*prodinf(k=0, (1+q(n)^(2*k+1))); 2^(1/4)/G(79)


CROSSREFS

Cf. A318733.
Sequence in context: A128213 A171716 A211788 * A016706 A138679 A179399
Adjacent sequences: A318729 A318730 A318731 * A318733 A318734 A318735


KEYWORD

nonn,cons


AUTHOR

Hugo Pfoertner, Sep 02 2018


STATUS

approved



