A308612 Moments of the ternary Cantor measure (numerators).

1, 1, 3, 5, 87, 31, 10215, 2675, 2030721, 3791353, 21043039755, 3617048975, 456510966890031, 66989072882759, 4380916942893971361, 8410761713598485675, 4355410489470724905492213, 2471070675390828342358441, 8410576873515817323688553597445

Offset: 0

Author

Alexander Riasanovsky, Jun 10 2019

Keywords

• nonn
• frac

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 numerators 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

Crossrefs

Cf. A308613 (denominators).

Programs

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[Numerator[a[i]], {i, 0, 19}] (* Amiram Eldar, Aug 03 2019 *)

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.numerator() for p in am(15, (1/2, 0, 1/2))]

Extensions

More terms from Amiram Eldar, Aug 03 2019