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A202300 Decimal expansion of the real root of x^3 + 2x^2 + 10x - 20. 1
1, 3, 6, 8, 8, 0, 8, 1, 0, 7, 8, 2, 1, 3, 7, 2, 6, 3, 5, 2, 2, 7, 4, 1, 4, 3, 3, 0, 0, 2, 1, 3, 2, 5, 5, 3, 9, 5, 4, 2, 4, 3, 5, 5, 4, 1, 4, 8, 7, 5, 3, 6, 5, 3, 0, 7, 9, 3, 7, 1, 2, 6, 9, 0, 2, 1, 8, 2, 6, 3, 1, 4, 7, 4, 1, 9, 6, 8, 8, 3, 8, 1, 9, 6, 9, 3, 9, 8, 8, 9 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

There is a small typo in Posamentier & Lehmann (2007): this number is given as approximately 1.3688081075 rather than 1.3688081078, a mistake that can't be justified by rounding rather than truncating nor a loss of machine precision. - Alonso del Arte, Mar 24 2012

Perhaps the reason for the mistake is that the authors got the correct answer mixed up with Fibonacci's answer, which, though wrong, was very good for the time: 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6 = 1.36880810785322... But apparently they truncated at the first 5 and left out the 8 before that 5. - Alonso del Arte, Jun 09 2014

The complex roots are -1.68440405391... +- 3.43133135... * i. - Alonso del Arte, Jun 21 2014

REFERENCES

John Derbyshire, Unknown Quantity: A Real and Imaginary History of Algebra. Washington, DC: Joseph Henry Press (2006): 69-70.

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 63-64.

Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers. New York: Prometheus Books (2007) p. 21.

LINKS

Table of n, a(n) for n=1..90.

Ezra Brown and Jason C. Brunson, Fibonacci's forgotten number

Stanislaw Glushkov, On approximation methods of Leonardo Fibonacci, Historia Mathematica 3 (1976), pp. 291-296.

Wolfram|Alpha, real root of x^3 + 2x^2 + 10x - 20 = 0

FORMULA

x = (2*sqrt(3930)/9 - 352/27)^(1/3) + (2*sqrt(3930)/9 + 352/27)^(1/3) - 2/3;

x = (1/3)*(-2 - 13 * 2^(2/3)/(176 + 3*sqrt(3930))^(1/3) + (2*(176 + 3*sqrt(3930)))^(1/3)).

The first formula comes from Posamentier & Lehmann (2007), the second from Wolfram|Alpha. - Alonso del Arte, Mar 24 2012

EXAMPLE

x = 1.36880810782137263522741433002132553954243554148753653...

MATHEMATICA

RealDigits[x /. FindRoot[x^3 + 2x^2 + 10x - 20 == 0, {x, 1.4}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Feb 27 2013 *)

PROG

(PARI) real(polroots(x^3+2*x^2+10*x-20)[1])

(PARI) polrootsreal(x^3+2*x^2+10*x-20)[1] \\ Charles R Greathouse IV, Jan 05 2016

CROSSREFS

Cf. A159990, A243629, A244467.

Sequence in context: A248760 A011261 A104541 * A244467 A200590 A036265

Adjacent sequences:  A202297 A202298 A202299 * A202301 A202302 A202303

KEYWORD

nonn,cons

AUTHOR

Charles R Greathouse IV, Jan 11 2012

STATUS

approved

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)