

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
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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): 6970.
Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 6364.
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. 291296.
WolframAlpha, 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 WolframAlpha.  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*x20)[1])
(PARI) polrootsreal(x^3+2*x^2+10*x20)[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



