A095844 Numerator of the integral of the n-th power of the Cantor function.

1, 1, 3, 1, 33, 5, 75, 611, 97653, 83057, 22018179, 9625216, 20894487717, 93120706729, 411117020063871, 297434062421057, 6650181371241300777, 6082551300359191981, 2198073713661546055399083

Offset

0,3

Links

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

Harold G. Diamond and Bruce Reznick, Problem 10621, Problems and Solutions, The American Mathematical Monthly, Vol. 104, No. 9 (1997), p. 870; Cantor's Singular Moments, Solutions to Problem 10621 by Kenneth F. Andersen and Omran Kouba, ibid., Vol. 106, No. 2 (1999), pp. 175-176.

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

J(n) = (2 + Sum_{k=1..n-1} binomial(n,k) * J(k))/(3*2^n-2) (Diamond and Reznick, 1997). - Amiram Eldar, Jan 26 2024

Example

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

Maple

seq(numer(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..18); # Emeric Deutsch, Feb 22 2005

Mathematica

a[n_] := Numerator[ 1/(n+1) - Sum[Binomial[n, 2 k]*Floor[2^(2k - 1) - 1]*BernoulliB[2k]/Floor[(3*2^(2k - 1) - 1)*(n - 2k + 1)], {k, 1, Floor[n/2]}]]; Table[a[n], {n, 0, 18}] (* 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}]); Numerator[Array[f, 20, 0]] (* Amiram Eldar, Jan 26 2024 *)

Crossrefs

Cf. A095845 (denominators).

Sequence in context: A016481 A303818 A047815 * A113110 A317363 A190964

Adjacent sequences: A095841 A095842 A095843 * A095845 A095846 A095847

Keyword

nonn,frac

Author

Eric W. Weisstein, Jun 08 2004

Status

approved