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 A318733 Decimal expansion of the nontrivial real solution to x^6 + x^5 - x^3 - x^2 - x + 1 = 0. 1
 5, 7, 6, 4, 7, 1, 4, 2, 9, 6, 1, 9, 5, 5, 0, 6, 1, 0, 4, 8, 6, 3, 5, 4, 4, 0, 0, 1, 7, 7, 5, 7, 8, 5, 1, 7, 4, 7, 7, 3, 4, 2, 1, 8, 2, 1, 6, 1, 4, 7, 9, 0, 4, 9, 5, 3, 1, 2, 0, 0, 5, 8, 8, 4, 2, 6, 1, 1, 8, 7, 9, 3, 3, 9, 2, 6, 3 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The second 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^5 - x^3 - x^2 - x + 1 = 0 can be expressed in terms of radicals." The polynomial includes a trivial factor, i.e., x^6 + x^5 - x^3 - x^2 - x + 1 = (x - 1) * (x^5 + 2*x^4 + 2*x^3 + x^2 - 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 Table of n, a(n) for n=0..76. 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 Expressed in radicals, the number is (1/20)*4^(4/5)*((215*sqrt(5)*sqrt(235 + 94*sqrt(5)) - 10575 - 5405*sqrt(5) + 597*sqrt(235 + 94*sqrt(5)))/sqrt(235 + 94*sqrt(5)))^(1/5) - (329*sqrt(5)/sqrt(235 + 94*sqrt(5)) - 57*sqrt(5) + 9*sqrt(235 + 94*sqrt(5)) - 89)*4^(3/5)/(20*((215*sqrt(5)*sqrt(235 + 94*sqrt(5)) - 10575 - 5405*sqrt(5) + 597*sqrt(235 + 94*sqrt(5)))/sqrt(235 + 94*sqrt(5)))^(3/5)) - (47*sqrt(5)/sqrt(235 + 94*sqrt(5)) + 23*sqrt(5) - 3*sqrt(235 + 94*sqrt(5)) - 3)* 4^(2/5)/(20*((215*sqrt(5)*sqrt(235 + 94*sqrt(5)) - 10575 - 5405*sqrt(5) + 597*sqrt(235 + 94*sqrt(5)))/sqrt(235 + 94*sqrt(5)))^(2/5)) + (-1 + 2*sqrt(5))*4^(1/5)/(5*((215*sqrt(5)*sqrt(235 + 94*sqrt(5)) - 10575 - 5405*sqrt(5) + 597*sqrt(235 + 94*sqrt(5)))/sqrt(235 + 94*sqrt(5)))^(1/5)) - 2/5. - Robert Israel, Sep 04 2018 Equals 2^(1/4) / G(47), 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)). - Hugo Pfoertner, Sep 15 2018 EXAMPLE 0.5764714296195506104863544001775785174773421821614790... PROG (PARI) p(x)=x^5+2*x^4+2*x^3+x^2-1; solve(x=0.3, 0.7, p(x)) CROSSREFS Cf. A318732. Sequence in context: A196615 A305200 A198730 * A195444 A114603 A348731 Adjacent sequences: A318730 A318731 A318732 * A318734 A318735 A318736 KEYWORD nonn,cons AUTHOR Hugo Pfoertner, Sep 02 2018 STATUS approved

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Last modified December 11 17:32 EST 2023. Contains 367727 sequences. (Running on oeis4.)