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, 119935974414957427604889850, 3156209853025195463286575

Offset: 0

Author

Eric W. Weisstein, Jun 08 2004

Keywords

• nonn
• frac

Examples

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

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.

Crossrefs

Cf. A095844 (numerators).

Programs

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 *)

Formula

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