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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 A100554

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 June 19 12:57 EDT 2019. Contains 324222 sequences. (Running on oeis4.)