A272343 -id:A272343 - cp's OEIS frontend

A272755 Numerators of the Fabius function F(1/2^n).

1, 1, 5, 1, 143, 19, 1153, 583, 1616353, 132809, 134926369, 46840699, 67545496213157, 4068990560161, 411124285571171, 1204567303451311, 73419800947733963069, 4146897304424408411, 86773346866163284480799923, 18814360006695807527868793, 539741515875650532056045666422369

Offset: 0

Vladimir Reshetnikov, May 05 2016

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.

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

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.

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, Probability Theory and Related Fields, June 1966, Volume 5, Issue 2, pp 173-174.
Jan Kristian Haugland, Evaluating the Fabius function, arXiv:1609.07999 [math.GM], 23 Sep 2016.
Wikipedia, Fabius function

Cf. A272757 (denominators), A272343.

c[0] = 1; c[n_] := c[n] = Sum[(-1)^k c[n - k]/(2 k + 1)!, {k, 1, n}] / (4^n - 1); Numerator@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 *)

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

A272757 Denominators of the Fabius function F(1/2^n).

Original entry on oeis.org

1, 2, 72, 288, 2073600, 33177600, 561842749440, 179789679820800, 704200217922109440000, 180275255788060016640000, 1246394851358539387238350848000, 6381541638955721662660356341760000, 292214732887898713986916575925267070976000000

Offset: 0

Vladimir Reshetnikov, May 05 2016

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)

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

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

Cf. A272755 (numerators), A272343.

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 *)

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

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

Original entry on oeis.org

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

Offset: 2

Vladimir Reshetnikov, Oct 14 2016

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

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.

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

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

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}]

A277430 Denominators of the Fabius function F(3/2^n).

Original entry on oeis.org

72, 288, 2073600, 33177600, 2809213747200, 179789679820800, 704200217922109440000, 180275255788060016640000, 6231974256792696936191754240000, 6381541638955721662660356341760000, 292214732887898713986916575925267070976000000, 1196911545908833132490410294989893922717696000000

Offset: 2

Vladimir Reshetnikov, Oct 14 2016

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

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.

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

Cf. A277429 (numerators), A272755, A272757, A272343.

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[Denominator[f[3/2^n]], {n, 2, 20}]

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A272755 Numerators of the Fabius function F(1/2^n).

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A272757 Denominators of the Fabius function F(1/2^n).

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

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A277430 Denominators of the Fabius function F(3/2^n).

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