A106533 The rumor constant: decimal expansion of the number x defined by x*e^(2 - 2*x) = 1.

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

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

L. Bayon, P. Fortuny Ayuso, J. M. Grau, A. M. Oller-Marcen, M. M. Ruiz, The Best-or-Worst and the Postdoc problems, arXiv:1706.07185 [math.PR], 2017.

L. Bayon, J. Grau, A. M. Oller-Marcen, M. Ruiz, P. M. Suarez, A variant of the Secretary Problem: the Best or the Worst, arXiv preprint arXiv:1603.03928 [math.PR], 2016.

D. J. Daley and D. G. Kendall, Stochastic rumours, Journal of the Institute of Mathematics and Its Applications 1:42-55, 1965.

Steven R. Finch, Errata and Addenda to Mathematical Constants.

José María Grau Ribas, A New Proof and Extension of the Odds-Theorem, arXiv:1812.09255 [math.PR], 2018.

José María Grau Ribas, An extension of the Last-Success-Problem, Statistics & Probability Letters (2020) Vol. 156, 108591.

Brian Hayes, Rumours and Errours, American Scientist, Vol. 93, Nbr. 3, 2005, pp. 207-211.

Formula

Solution to x*exp(2 - 2*x) = 1 with x not equal to 1.

Equals -1/2*LambertW(-2*exp(-2)). - Vladeta Jovovic, May 30 2005

Constant c satisfies: exp(c*x)/(1-2*c) = Sum_{n>=0} (x + 2*n)^n * exp(-2*n)/n!. - Paul D. Hanna, Mar 12 2016

Equals (2-A256500)/2. - Miko Labalan, Dec 18 2024

Example

c = 0.20318786997997995383847906206241987910549878759057031750024774...

Mathematica

RealDigits[ -ProductLog[ -2/E^2]/2, 10, 111][[1]]

Prog

(PARI) solve(x=0, 0.5, x*exp(2-2*x)-1) \\ Michel Marcus, Mar 13 2016

Crossrefs

Cf. A007456, A129506, A217900, A217910, A256500.

Sequence in context: A241644 A241640 A158449 * A192421 A035223 A035184

Adjacent sequences: A106530 A106531 A106532 * A106534 A106535 A106536

Keyword

cons,nonn

Author

Robert G. Wilson v, May 03 2005

Status

approved