A307951 Decimal expansion of 1 - log(2)/log(-W(-2/e^2)), where W is Lambert's W function.

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

Offset: 1

Author

Charles R Greathouse IV, May 07 2019

Keywords

• nonn
• cons

Comments

Chang shows that a constant population of n individuals, with ancestors selected uniformly at random, converges in probability to a state where every individual leaves either no current ancestors or else is a common ancestor of all present individuals after k*log_2(n) generations, where k is this constant (see Theorem 2 in Chang link for precise statement).

Examples

			1.769755495564801280059561457905786683522251513088978630155101689614415...
A population of 1000 is expected to have identical ancestors after around k*log_2(1000) = 17.6... generations.
A population of a million is expected to have identical ancestors after around k*log_2(10^6) = 35.2... generations.
A population of a billion is expected to have identical ancestors after around k*log_2(10^9) = 52.9... generations.
A population of a trillion is expected to have identical ancestors after around k*log_2(10^12) = 70.5... generations.
		

Links

• Joseph T. Chang, Recent common ancestors of all present-day individuals, Advances in Applied Probability Vol. 31, No. 4 (Dec., 1999), pp. 1002-1026.
• James Grime and Brady Haran, EVERY baby is a ROYAL baby, Numberphile video (2019).
• Douglas L. T. Rohde, Steve Olson, and Joseph T. Chang, Modelling the recent common ancestry of all living humans, Nature Vol. 431, No. 7008 (Sep. 2004), pp. 562-566.

Crossrefs

Cf. A106533, A226775.

Programs

Mathematica

RealDigits[1 - Log[2]/Log[-ProductLog[-2/E^2]], 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)

PARI

1 - log(2)/log(-lambertw(-2/exp(2))) \\ Charles R Greathouse IV, Jan 24 2025