0,2

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. It assumes rational values at dyadic rationals.

From Juan Arias-de-Reyna, Jun 08 2017: (Start)

It is true that n! divides a(n) for all n? This is true for the first 200 terms.

If this is true A272755, the sequence of numerators of F(2^(-n)) is also the sequence of numerators of the half moments of Rvachëv function. (Cf. A288161). (End)

Rvachev V. L., Rvachev V. A., Non-classical methods of the approximation theory in boundary value problems, Naukova Dumka, Kiev (1979) (in Russian), pages 117-125.

Vincenzo Librandi, Table of n, a(n) for n = 0..64

Juan Arias de Reyna, An infinitely differentiable function with compact support: Definition and properties, arXiv:1702.05442 [math.CA], 2017.

Juan Arias de Reyna, On the arithmetic of Fabius function, arXiv:1702.06487 [math.NT], 2017.

Yuri Dimitrov, G. A. Edgar, Solutions of Self-differential Functional Equations

G. A. Edgar, Examples of self differential functions

J. Fabius, A probabilistic example of a nowhere analytic C^infty-function, Z. Wahrscheinlichkeitstheorie verw. Gebiete (1966) 5: 173.

Jan Kristian Haugland, Evaluating the Fabius function, arXiv:1609.07999 [math.GM], 23 Sep 2016.

Wikipedia, Fabius function

Recurrence: d(0) = 1, d(n) = (1/(n+1)! + Sum_{k=1..n-1} (2^(k*(k-1)/2)/(n-k+1)!)*d(k))/((2^n-1)*2^(n*(n-1)/2)), where d(n) = A272755(n)/A272757(n). - Vladimir Reshetnikov, Feb 27 2017

A272755/A272757 = 1/1, 1/2, 5/72, 1/288, 143/2073600, 19/33177600, 1153/561842749440, 583/179789679820800, ...

c[0] = 1; c[n_] := c[n] = Sum[(-1)^k c[n - k]/(2 k + 1)!, {k, 1, n}] / (4^n - 1); Denominator@Table[Sum[c[k] (-1)^k / (n - 2 k)!, {k, 0, n/2}] / 2^((n + 1) n/2), {n, 0, 15}] (* Vladimir Reshetnikov, Oct 16 2016 *)

Cf. A272755 (numerators), A272343.

Sequence in context: A217842 A383350 A053318 * A187707 A163274 A030993

Adjacent sequences: A272754 A272755 A272756 * A272758 A272759 A272760

nonn,frac

Vladimir Reshetnikov, May 05 2016

approved