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A086254 Decimal expansion of Feller's beta coin-tossing constant.

%I A086254 #38 Feb 16 2025 08:32:50
%S A086254 1,2,3,6,8,3,9,8,4,4,6,3,8,7,8,5,1,0,1,8,9,0,6,6,0,8,7,6,1,4,2,1,2,3,
%T A086254 2,5,2,2,1,1,1,7,6,6,2,1,2,3,5,8,8,5,8,7,3,7,1,0,7,1,6,7,2,6,7,1,5,9,
%U A086254 0,4,2,7,4,0,0,9,2,5,8,8,1,9,1,0,7,7,8,3,8,2,6,1,3,0,6,3,9,9,3,5,7,5,9,1
%N A086254 Decimal expansion of Feller's beta coin-tossing constant.
%D A086254 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.11 Feller's coin tossing constants, p. 339.
%H A086254 G. C. Greubel, <a href="/A086254/b086254.txt">Table of n, a(n) for n = 1..10000</a>
%H A086254 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Run.html">Run</a>.
%H A086254 Wikipedia, <a href="https://en.wikipedia.org/wiki/Feller%27s_coin-tossing_constants">Feller's coin-tossing constants</a>.
%F A086254 Equals (22 + (847 - 33*sqrt(33))^(1/3) + (11*(77 + 3*sqrt(33)))^(1/3))/33. - _Vaclav Kotesovec_, Oct 14 2018
%F A086254 Positive real root of 11*x^3 - 22*x^2 + 12*x - 2. - _Peter Luschny_, Oct 14 2018
%F A086254 Equals 2/(5 - A058265^2). - _Jon Maiga_, Nov 25 2018
%e A086254 1.2368398446387851018906608761421232....
%p A086254 evalf[120](solve(11*x^3-22*x^2+12*x-2=0,x)[1]); # _Muniru A Asiru_, Nov 25 2018
%t A086254 alpha = Root[1-x+(x/2)^4, x, 1]; beta = (2-alpha)/(4-3*alpha); RealDigits[beta, 10, 102] // First (* _Jean-François Alcover_, Jun 03 2014 *)
%o A086254 (PARI) default(realprecision, 100); (22 + (847 - 33*sqrt(33))^(1/3) + (11*(77 + 3*sqrt(33)))^(1/3))/33 \\ _G. C. Greubel_, Nov 25 2018
%o A086254 (Magma) SetDefaultRealField(RealField(100)); (22 + (847 - 33*Sqrt(33))^(1/3) + (11*(77 + 3*Sqrt(33)))^(1/3))/33; // _G. C. Greubel_, Nov 25 2018
%o A086254 (Sage) numerical_approx((22 + (847 - 33*sqrt(33))^(1/3) + (11*(77 + 3*sqrt(33)))^(1/3))/33, digits=100) # _G. C. Greubel_, Nov 25 2018
%Y A086254 Cf. A086253, A058265.
%K A086254 nonn,cons
%O A086254 1,2
%A A086254 _Eric W. Weisstein_, Jul 13 2003