A095845 -id:A095845 - cp's OEIS frontend

A047816 Erroneous version of A095845.

1, 2, 10, 5, 230, 46, 35

Offset: 0

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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, 53388901948383223161199, 31122898898234908646386438959

Offset: 0

Eric W. Weisstein, Jun 08 2004

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

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.

Cf. A095845 (denominators).

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

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 *)

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
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

A159009 Numerator of the integral of x^n times the Cantor function, from 0 to 1.

Original entry on oeis.org

1, 5, 11, 233, 97, 36377, 10637, 8885119, 18040327, 107868664309, 19821442673, 2657527033463249, 412093696402361, 28353905269136197727, 57058882710461852501, 30872757660805358101602571

Offset: 0

Simon Tatham (anakin(AT)pobox.com), Apr 02 2009

			I(0) is obviously 1/2 by symmetry.

A095844/A095845 give the integrals of powers of the Cantor function itself.

A159010 gives the corresponding denominators. [From Simon Tatham (anakin(AT)pobox.com), Apr 02 2009]

for n from 0 to 20 do CI[n] := 1/(2*(n+1)) + 1/(2*(3^(n+1)-1)) * add(binomial(n,i)*2^(n-i)*CI[i],i=0..n-1); end do;

I(n) = 1/(2*(n+1)) + 1/(2*3^(n+1)-1) * sum_{i=0}{n-1} (n choose i) 2^(n-i) I(i)

A159010 Denominator of the integral of x^n times the Cantor function, from 0 to 1.

Original entry on oeis.org

2, 16, 48, 1280, 640, 279552, 93184, 87326720, 196485120, 1289117040640, 257823408128, 37368456004239360, 6228076000706560, 458287510968422367232, 982044666360905072640, 563650690404417328113516544

Offset: 0

Simon Tatham (anakin(AT)pobox.com), Apr 02 2009

			I(0) is obviously 1/2 by symmetry.

A159009 gives the corresponding numerators. A095844/A095845 give the integrals of powers of the Cantor function itself.

I(n) = 1/(2*(n+1)) + 1/(2*3^(n+1)-1) * sum_{i=0}{n-1} (n choose i) 2^(n-i) I(i)

A113223 Decimal expansion of Integral_{t=0..1} F(t)^F(t) dt, where F(t) is the Cantor function.

Original entry on oeis.org

7, 5, 0, 3, 8, 7, 0, 8, 2, 1, 6, 4, 1, 9, 8, 5, 1, 4, 7, 9, 5, 3, 1, 9, 7, 3, 6, 8, 0, 9, 3, 2, 9, 3, 3, 6, 9, 8, 6, 8, 1, 0, 3, 0, 2, 1, 0, 9, 4, 4, 4, 7, 7, 9, 8, 1, 8, 2, 2, 9, 6, 1, 1, 7, 1, 1, 9, 6, 5, 8, 3, 2, 4, 2, 5, 8, 7, 0, 4, 6, 9, 9, 8, 8, 4, 4, 8, 5, 6, 6, 6, 8, 9, 4, 2, 6, 4, 9, 8, 9, 7, 1

Offset: 0

Eric W. Weisstein, Oct 18 2005

			0.750387082164198514795...

Eric Weisstein's World of Mathematics, Cantor Function

Cf. A095844, A095845.

A369501 Decimal expansion of the integral of the reciprocal of the Cantor function.

Original entry on oeis.org

3, 3, 6, 4, 6, 5, 0, 7, 2, 8, 1, 0, 0, 9, 2, 5, 1, 6, 0, 8, 3, 8, 9, 3, 4, 9, 6, 2, 8, 9

Offset: 1

Amiram Eldar, Jan 25 2024

			3.36465072810092516083893496289...

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.
Steven Finch, Cantor-solus and Cantor-multus Distributions, arXiv:2003.09458 [math.CO], 2020.
Russell A. Gordon, Some Integrals Involving the Cantor Function, The American Mathematical Monthly, Vol. 116, No. 3 (2009), pp. 218-227; alternative link.
Helmut Prodinger, On Cantor's singular moments, Southwest Journal of Pure and Applied Mathematics, Vol. 2000, Issue 1 (July 2000), pp. 27-29; arXiv preprint, arXiv:math/9904072 [math.CO], 1999.
Helmut Prodinger, Digits and beyond, in: B. Chauvin, P. Flajolet, D. Gardy, and A. Mokkadem (eds.), Mathematics and Computer Science II: Algorithms, Trees, Combinatorics and Probabilities, Birkhäuser, Basel (2012), pp. 355-377.
Eric Weisstein's World of Mathematics, Cantor Function.
Wikipedia, Cantor function.

Cf. A001008, A002805, A095844, A095845.

Equals Integral_{x=0..1} (1/c(x)) dx, where c(x) is the Cantor function.
Equals Sum_{k>=0} Integral_{x=0..1} c(x)^k dx = Sum_{k>=0} A095844(k)/A095845(k) (Javier Duoandikoetx, in "Cantor's Singular Moments", 1999).
Equals -1/3 + (2/3) * Sum_{k>=1} (2/3)^k * H(2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Prodinger, 2000).

cp's OEIS Frontend

A047816 Erroneous version of A095845.

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

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A159009 Numerator of the integral of x^n times the Cantor function, from 0 to 1.

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A159010 Denominator of the integral of x^n times the Cantor function, from 0 to 1.

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A113223 Decimal expansion of Integral_{t=0..1} F(t)^F(t) dt, where F(t) is the Cantor function.

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A369501 Decimal expansion of the integral of the reciprocal of the Cantor function.

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