A308613 Moments of the ternary Cantor measure (denominators).

1, 2, 8, 16, 320, 128, 46592, 13312, 10915840, 21831680, 128911704064, 23438491648, 3114038000353280, 479082769285120, 32734822212030169088, 65469644424060338176, 35228168150276083007094784, 20722451853103578239467520, 72984567358962659964369885986816

Offset

0,2

Comments

The ternary Cantor measure, defined many ways, is the unique Borel measure mu on the unit interval [0,1] satisfying the following recurrence relation for any measurable set E: mu(E) = mu(phi_0(E))/2 + mu(phi_2(E))/2. Here, for j in {0,1,2}, phi_j: [0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/3. For any nonnegative integer k, we define the k-th moment to be I(k) to be the integral of x^k with respect to mu. The described sequence I(0), I(1), I(2), ... is rational and this sequence a(0), a(1), a(2), ... is the sequence of denominators of I(0), I(1), I(2), ....

For the purpose of computing I(k), we note the following recurrence relation: I(0) = 1 and for all positive k, I(k) = (1/(3^k-1))*((1/2) * Sum_{j=0..k-1} binomial(k, j) + (1/2) * Sum_{j=0..k-1} binomial(k, j) * 2^(k-j) * I(j)).

More generally, for any N-dimensional nonnegative vector alpha = (alpha_0, ..., alpha_{N-1}) whose entries sum to 1, there exists a unique Borel measure mu = mu^{alpha} on [0,1] so that for any measurable set E, the following identity holds: mu(E) = Sum_{k=0..N-1} alpha_k * mu(phi_k(E)). Here, for j in {0, 1, ..., N-1}, phi_j: [0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/N. Defining I(k) to be the integral of x^k with respect to mu, the following recurrence relation holds: I(0) = 1 and for all positive k, I(k) = (1/(N^k-1)) * Sum_{n=0..N-1} alpha_n * Sum_{j=0..k-1} binomial(k, j) * n^(k-j)*I(j).

Links

Michael De Vlieger, Table of n, a(n) for n = 0..97

David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Michael G. Rose, Expectations on fractal sets, Applied Mathematics and Computation, Vol. 220 (2013), pp. 695-721, alternative link.

Steven N. Harding, Alexander W. N. Riasanovsky, Moments of the weighted Cantor measures, arXiv:1908.05358 [math.FA], 2019.

Math Stack Exchange, Integrating f(x) = x for x in C, the Cantor set, with respect to a certain measure

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, j]*2^(n - j - 1)*a[j], {j, 0, n - 1}]/(3^n - 1); Table[Denominator[a[i]], {i, 0, 19}] (* Amiram Eldar, Aug 03 2019 *)

Prog

(Sage)
def am(m, alpha):
    N = len(alpha)
    am = [1]
    for a in [1..m]:
        mm = 0
        for k in [0..N-1]:
            for r in [0..a-1]:
                mm += alpha[k]*binomial(a, r)*k^(a-r)*am[r]
        mm /= (N^a-1)
        am.append(mm)
    return am
[p.denominator() for p in am(15, (1/2, 0, 1/2))]

Crossrefs

Cf. A308612 (numerators).

Sequence in context: A101059 A191999 A101658 * A006573 A009662 A279243

Adjacent sequences: A308610 A308611 A308612 * A308614 A308615 A308616

Keyword

nonn,frac

Author

Alexander Riasanovsky, Jun 10 2019

Extensions

More terms from Amiram Eldar, Aug 03 2019

Status

approved