A308613 - OEIS

%I #40 Nov 19 2023 15:56:08

%S 1,2,8,16,320,128,46592,13312,10915840,21831680,128911704064,

%T 23438491648,3114038000353280,479082769285120,32734822212030169088,

%U 65469644424060338176,35228168150276083007094784,20722451853103578239467520,72984567358962659964369885986816

%N Moments of the ternary Cantor measure (denominators).

%C 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), ....

%C 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)).

%C 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).

%H Michael De Vlieger, <a href="/A308613/b308613.txt">Table of n, a(n) for n = 0..97</a>

%H David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Michael G. Rose, <a href="https://doi.org/10.1016/j.amc.2013.06.078">Expectations on fractal sets</a>, Applied Mathematics and Computation, Vol. 220 (2013), pp. 695-721, <a href="https://www.davidhbailey.com/dhbpapers/fracboxes.pdf">alternative link</a>.

%H Steven N. Harding, Alexander W. N. Riasanovsky, <a href="https://arxiv.org/abs/1908.05358">Moments of the weighted Cantor measures</a>, arXiv:1908.05358 [math.FA], 2019.

%H Math Stack Exchange, <a href="https://math.stackexchange.com/q/57721">Integrating f(x) = x for x in C, the Cantor set, with respect to a certain measure</a>

%t 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 *)

%o (Sage)

%o def am(m, alpha):

%o     N = len(alpha)

%o     am = [1]

%o     for a in [1..m]:

%o         mm = 0

%o         for k in [0..N-1]:

%o             for r in [0..a-1]:

%o                 mm += alpha[k]*binomial(a, r)*k^(a-r)*am[r]

%o         mm /= (N^a-1)

%o         am.append(mm)

%o     return am

%o [p.denominator() for p in am(15, (1/2, 0, 1/2))]

%Y Cf. A308612 (numerators).

%K nonn,frac

%O 0,2

%A _Alexander Riasanovsky_, Jun 10 2019

%E More terms from _Amiram Eldar_, Aug 03 2019