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%I #23 May 27 2023 04:16:49
%S 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,
%T 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,
%U 9,5,5,6,4,3,5,4,1,5,8,4,7,2,5,5,4,9,4
%N Decimal expansion of F(1/3), where F(x) is the Fabius function.
%C 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.
%C 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.
%H Vladimir Reshetnikov, <a href="/A272343/b272343.txt">Table of n, a(n) for n = 0..189</a>
%H Yuri Dimitrov, G. A. Edgar, <a href="http://projecteuclid.org/euclid.rae/1184700035">Solutions of Self-differential Functional Equations</a>, Real Anal. Exchange 32 (2006), no. 1, 29--54.
%H Gerald A. Edgar, <a href="http://people.math.osu.edu/edgar.2/selfdiff/">Examples of self differential functions</a>.
%H Jaap Fabius, <a href="http://dx.doi.org/10.1007/BF00536652">A probabilistic example of a nowhere analytic C^infty-function</a>, Probability Theory and Related Fields, Volume 5, Issue 2 (June 1966), pp. 173-174.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fabius_function">Fabius function</a>.
%e 0.1801651148014819069557334359310241228679078...
%t RealDigits[ResourceFunction["FabiusF"][1/3], 10, 120][[1]] (* _Amiram Eldar_, May 27 2023 *)
%K nonn,cons,hard
%O 0,2
%A _Vladimir Reshetnikov_, Apr 26 2016