A272343 Decimal expansion of F(1/3), where F(x) is the Fabius function.

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

Offset: 0

Author

Vladimir Reshetnikov, Apr 26 2016

Keywords

• nonn
• cons
• hard

Comments

The Fabius function F(x) is the smooth monotone increasing function on [0, 1] satisfying F(0) = 0, F(1) = 1, F'(x) = 2*F(2*x) for 0 < x < 1/2, F'(x) = 2*F(2*(1-x)) for 1/2 < x < 1. It is infinitely differentiable at every point in the interval, but is nowhere analytic.

The numeric value of F(1/3) was calculated using Wynn's epsilon method applied to a sequence of piecewise polynomial approximations to the Fabius function.

Examples

			0.1801651148014819069557334359310241228679078...
		

Links

• Vladimir Reshetnikov, Table of n, a(n) for n = 0..189
• Yuri Dimitrov, G. A. Edgar, Solutions of Self-differential Functional Equations, Real Anal. Exchange 32 (2006), no. 1, 29--54.
• Gerald A. Edgar, Examples of self differential functions.
• Jaap Fabius, A probabilistic example of a nowhere analytic C^infty-function, Probability Theory and Related Fields, Volume 5, Issue 2 (June 1966), pp. 173-174.
• Wikipedia, Fabius function.

Programs

Mathematica

RealDigits[ResourceFunction["FabiusF"][1/3], 10, 120][[1]] (* Amiram Eldar, May 27 2023 *)