Congruence Properties of the Function that Counts Compositions into Powers of 2

Let θ(n) denote the number of compositions (ordered partitions) of a positive integer n into powers of 2. It appears that the function θ(n) satisfies many congruences modulo 2 . For example, for every integer a there exists (as k tends to infinity) the limit of θ(2 + a) in the 2−adic topology. The parity of θ(n) obeys a simple rule. In this paper we extend this result to higher powers of 2. In particular, we prove that for each positive integer N there exists a finite table which lists all the possible cases of this sequence modulo 2 . One of our main results claims that θ(n) is divisible by 2 for almost all n, however large the value of N is. The author gratefully acknowledges support from the Austrian Science Fund (FWF) under the project Nr. P20847-N18.