A277429 Numerators of the Fabius function F(3/2^n).

67, 73, 46657, 25219, 29407171, 10997359, 109661317247, 31733679209, 558462830097043, 132566737763827, 646476041042787542443, 130499244418507180561, 2411172049639892707896547, 424191560077453917728503, 84883189962706557116984038531, 172244289373664036915914887721

Offset

2,1

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. It assumes rational values at dyadic rationals.

References

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.

Links

Table of n, a(n) for n=2..17.

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, Probability Theory and Related Fields, June 1966, Volume 5, Issue 2, pp 173-174.

Wikipedia, Fabius function

Example

A277429/A277430 = 67/72, 73/288, 46657/2073600, 25219/33177600, 29407171/2809213747200, ... (starting from n = 2)

Mathematica

c[0] = 1;
c[k_] := c[k] = Sum[((-1)^(k - r) c[r])/(1 + 2 k - 2 r)!, {r, 0, k - 1}]/(4^k - 1);
t[n_] := Mod[2 n + Sum[(-1)^Binomial[n, k], {k, 1, n}], 3];
f[x_] := Module[{k = Numerator[x], n = Log2[Denominator[x]]}, Sum[((-1)^(q + t[p - 1]) 2^(-(n - 1) n/2) (1/2 - p + k)^(n - 2 q) c[q])/(4^q (n - 2 q)!), {p, 1, k}, {q, 0, n/2}]];
Table[Numerator[f[3/2^n]], {n, 2, 20}]

Crossrefs

Cf. A277430 (denominators), A272755, A272757, A272343.

Sequence in context: A241897 A225626 A345214 * A039432 A043255 A044035

Adjacent sequences: A277426 A277427 A277428 * A277430 A277431 A277432

Keyword

nonn,frac

Author

Vladimir Reshetnikov, Oct 14 2016

Status

approved