cp's OEIS Frontend

A202300 Decimal expansion of the real root of x^3 + 2x^2 + 10x - 20.

%I A202300 #56 Oct 27 2023 10:31:43
%S A202300 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,
%T A202300 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,
%U A202300 6,3,1,4,7,4,1,9,6,8,8,3,8,1,9,6,9,3,9,8,8,9
%N A202300 Decimal expansion of the real root of x^3 + 2x^2 + 10x - 20.
%C A202300 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
%C A202300 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
%C A202300 The complex roots are -1.68440405391... +- 3.43133135... * i. - _Alonso del Arte_, Jun 21 2014
%C A202300 Fibonacci calculated this constant to six sexagesimal digits and proved that it was neither rational nor a square root of a rational. - _Charles R Greathouse IV_, Oct 21 2022
%D A202300 John Derbyshire, Unknown Quantity: A Real and Imaginary History of Algebra. Washington, DC: Joseph Henry Press (2006): 69-70.
%D A202300 Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 63-64.
%D A202300 Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers. New York: Prometheus Books (2007) p. 21.
%H A202300 Ezra Brown and Jason C. Brunson, <a href="http://www.math.vt.edu/people/brown/doc/fibo_number.pdf">Fibonacci's forgotten number</a> (<a href="https://web.archive.org/web/20160719194355/http://www.math.vt.edu/people/brown/doc/fibo_number.pdf">archived link</a>)
%H A202300 Stanislaw Glushkov, <a href="http://dx.doi.org/10.1016/0315-0860(76)90098-7">On approximation methods of Leonardo Fibonacci</a>, Historia Mathematica 3 (1976), pp. 291-296.
%H A202300 Wolfram|Alpha, <a href="http://www.wolframalpha.com/input/?i=real+root+of+x%5E3+%2B+2x%5E2+%2B+10x+-+20+%3D+0">real root of x^3 + 2x^2 + 10x - 20 = 0</a>
%H A202300 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F A202300 x = (2*sqrt(3930)/9 - 352/27)^(1/3) + (2*sqrt(3930)/9 + 352/27)^(1/3) - 2/3;
%F A202300 x = (1/3)*(-2 - 13 * 2^(2/3)/(176 + 3*sqrt(3930))^(1/3) + (2*(176 + 3*sqrt(3930)))^(1/3)).
%F A202300 The first formula comes from Posamentier & Lehmann (2007), the second from Wolfram|Alpha. - _Alonso del Arte_, Mar 24 2012
%e A202300 x = 1.36880810782137263522741433002132553954243554148753653...
%t A202300 RealDigits[x /. FindRoot[x^3 + 2x^2 + 10x - 20 == 0, {x, 1.4}, WorkingPrecision -> 120]][[1]] (* _Harvey P. Dale_, Feb 27 2013 *)
%o A202300 (PARI) real(polroots(x^3+2*x^2+10*x-20)[1])
%o A202300 (PARI) polrootsreal(x^3+2*x^2+10*x-20)[1] \\ _Charles R Greathouse IV_, Jan 05 2016
%Y A202300 Cf. A159990, A243629, A244467.
%K A202300 nonn,cons
%O A202300 1,2
%A A202300 _Charles R Greathouse IV_, Jan 11 2012