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 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 (list; constant; graph; refs; listen; history; text; internal format)
 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. 89-94 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 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 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^5-x^4+x^3-2*x^2+3*x-1; 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

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Last modified February 16 16:00 EST 2020. Contains 331961 sequences. (Running on oeis4.)