A086253 -id:A086253 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

Eric W. Weisstein, Jul 13 2003

			1.2368398446387851018906608761421232....

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.11 Feller's coin tossing constants, p. 339.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Run.
Wikipedia, Feller's coin-tossing constants.

Cf. A086253, A058265.

SetDefaultRealField(RealField(100)); (22 + (847 - 33*Sqrt(33))^(1/3) + (11*(77 + 3*Sqrt(33)))^(1/3))/33; // G. C. Greubel, Nov 25 2018

evalf[120](solve(11*x^3-22*x^2+12*x-2=0,x)[1]); # Muniru A Asiru, Nov 25 2018

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 *)

default(realprecision, 100); (22 + (847 - 33*sqrt(33))^(1/3) + (11*(77 + 3*sqrt(33)))^(1/3))/33 \\ G. C. Greubel, Nov 25 2018

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

Equals (22 + (847 - 33*sqrt(33))^(1/3) + (11*(77 + 3*sqrt(33)))^(1/3))/33. - Vaclav Kotesovec, Oct 14 2018
Positive real root of 11*x^3 - 22*x^2 + 12*x - 2. - Peter Luschny, Oct 14 2018
Equals 2/(5 - A058265^2). - Jon Maiga, Nov 25 2018

A244293 Decimal expansion of 3/2 - gamma / log(2), a coin tossing constant related to the asymptotic evaluation of the expected length of the longest run of consecutive heads.

Original entry on oeis.org

6, 6, 7, 2, 5, 3, 8, 2, 2, 7, 2, 3, 1, 3, 2, 8, 4, 9, 3, 5, 3, 5, 8, 2, 4, 8, 0, 5, 9, 1, 8, 8, 4, 4, 6, 4, 8, 3, 7, 5, 6, 8, 4, 6, 8, 9, 7, 3, 6, 7, 1, 8, 9, 8, 3, 6, 8, 5, 0, 1, 8, 0, 2, 4, 1, 5, 4, 1, 8, 6, 4, 8, 5, 5, 3, 9, 0, 4, 1, 5, 7, 7, 0, 9, 7, 9, 7, 3, 9, 9, 6, 5, 5, 5, 6, 9, 3, 6, 9, 5, 3, 2

Offset: 0

Jean-François Alcover, Jun 25 2014

			0.66725382272313284935358248...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.11 Feller Coin Tossing Constants, p. 340.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Run.

Cf. A086253, A086254, A143300 (gamma/log(2)).

SetDefaultRealField(RealField(100)); R:=RealField(); 3/2 - EulerGamma(R)/Log(2); // G. C. Greubel, Oct 13 2018

RealDigits[3/2 - EulerGamma/Log[2], 10, 102] // First

default(realprecision, 100); 3/2 - Euler/log(2) \\ G. C. Greubel, Oct 13 2018

Expected length ~ log(n)/log(2) - (3/2-gamma/log(2)).

cp's OEIS Frontend

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

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