A320084 Decimal expansion of solution of -x*log_2(x) - (1-x)*log_2(1-x) + (1-x)*log_2(3) = 1.

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

Offset

0,1

Comments

Two people playing "online matching pennies" can get as close as they want to this fraction of success.

References

Peter Winkler, Mathematical Mind-Benders, A K Peters, 2017, ISBN 978-1-56881-336-3 (the number is given as "about 0.8016" on page 18).

Links

Table of n, a(n) for n=0..106.

Olivier Gossner, Penélope Hernández, and Abraham Neyman, Online Matching Pennies, 2003, Discussion Paper Series dp316, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.

Example

0.81071037508476823739760530663472575783303383815695...

Maple

evalf(solve(-x*log[2](x)-(1-x)*log[2](1-x)+(1-x)*log[2](3)=1, x), 120); # Muniru A Asiru, Oct 05 2018

Mathematica

RealDigits[x /. FindRoot[-x*Log2[x] - (1 - x)*Log2[1 - x] + (1 - x)*Log2[3] - 1, {x, 4/5}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Mar 27 2026 *)

Prog

(PARI) solve(x=0.1, 0.9, -x*log(x) - (1-x)*log(1-x) + (1-x)*log(3) - log(2)); \\ Michel Marcus, Oct 06 2018

Crossrefs

Sequence in context: A194732 A217739 A110544 * A153855 A340065 A011437

Adjacent sequences: A320081 A320082 A320083 * A320085 A320086 A320087

Keyword

nonn,cons

Author

Jack Zhang, Oct 05 2018

Status

approved