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 A095845 Denominator of the integral of the n-th power of the Cantor function. 7
 1, 2, 10, 5, 230, 46, 874, 8740, 1673710, 1673710, 513828970, 256914485, 631290272542, 3156451362710, 15513958447719650, 12411166758175720, 305013731457236950790, 305013731457236950790 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 8 17:50 EDT 2024. Contains 375753 sequences. (Running on oeis4.)