RECIPROCALS OF BINARY POWER SERIES
Joshua Cooper, Dennis Eichhorn, Kevin O’Bryant · International Journal of Number Theory · 2006
If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a + b, where a ∈ A and b ∈ B, in an even number of ways. We compute the natural density of B for several specific sets A, including the Prouhet–Thue–Morse sequence, {0} ∪ {2n:n ∈ ℕ}, and random sets, and we also study the distribution of densities of B for finite sets A. This problem is motivated by Euler's observation that if A is the set of n that has an odd number of partitions, then B is the set of pentagonal numbers {n(3n + 1)/2:n ∈ ℤ}. We also elaborate the connection between this problem and the theory of de Bruijn sequences and linear shift registers.