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

0,2

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.

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.

Example

0.1801651148014819069557334359310241228679078...

Mathematica

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

Crossrefs

Sequence in context: A321107 A198117 A241215 * A011314 A021559 A167176

Adjacent sequences: A272340 A272341 A272342 * A272344 A272345 A272346

Keyword

nonn,cons,hard

Author

Vladimir Reshetnikov, Apr 26 2016

Status

approved