A095845 Denominator of the integral of the n-th power of the Cantor function.

1, 2, 10, 5, 230, 46, 874, 8740, 1673710, 1673710, 513828970, 256914485, 631290272542, 3156451362710, 15513958447719650, 12411166758175720, 305013731457236950790, 305013731457236950790

Offset

0,2

Links

Amiram Eldar, Table of n, a(n) for n = 0..117

E. A. Gorin and B. N. Kukushkin, Integrals related to the Cantor function, St. Petersburg Math. J., 15, 449-468, 2004.

Eric Weisstein's World of Mathematics, Cantor Function.

Formula

The integral, a rational number, is given by J(n) = 1/(n+1) - Sum_{k = 1..floor(n/2)} (binomial(n,2k)*(2^(2k-1)-1)*bernoulli(2k)/((3*2^(2k-1)-1)*(n-2k+1)]). - Emeric Deutsch, Feb 22 2005

Note that the Cantor function C(x) satisfies C(x) = C(3*x)/2 for x in [0,1/3], 1/2 for x in [1/3,2/3] and (1+C(3*x-2))/2 for x in [2/3,1]. Integrating both sides yields J(n) = (1 + Sum_{k=0..n-1} binomial(n,k)*J(k))/(3*2^n - 2) with J(0) = 1, where J(n) := Integral_{x=0..1} (C(x))^n dx. - Jianing Song, Nov 19 2023

Example

1, 1/2, 3/10, 1/5, 33/230, 5/46, 75/874, 611/8740, 97653/1673710, ...

Maple

seq(denom(1/(n+1)-sum(binomial(n, 2*k)*(2^(2*k-1)-1)*bernoulli(2*k)/(3*2^(2*k-1)-1)/(n-2*k+1), k = 1 .. floor(1/2*n))), n=1..17); # Emeric Deutsch, Feb 22 2005

Mathematica

a[n_] := Denominator[1/(n + 1) - Sum[(Binomial[n, 2*k]*Floor[2^(2*k - 1) - 1]*BernoulliB[2*k])/Floor[(3*2^(2*k - 1) - 1)*(-2*k + n + 1)], {k, 1, Floor[n/2]}]]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Oct 23 2012, after Emeric Deutsch *)
f[0] = 1; f[1] = 1/2; f[n_] := f[n] = (1/(3*2^n - 2))*(2 + Sum[Binomial[n, k]*f[k], {k, 1, n - 1}]); Denominator[Array[f, 20, 0]] (* Amiram Eldar, Jan 26 2024 *)

Crossrefs

Cf. A095844 (numerators).

Sequence in context: A276850 A033468 A047816 * A105801 A086064 A076374

Adjacent sequences: A095842 A095843 A095844 * A095846 A095847 A095848

Keyword

nonn,frac

Author

Eric W. Weisstein, Jun 08 2004

Status

approved