Generalization of the Rödseth-Gupta theorem on binary partitions (Q1881791)

Let \(b(n)\) be the number of nonnegative solutions of the equation \(n=m_s2^s+m_{s-1}2^{s-1}+\dots+m_0\), where \(n\) is a positive integer. The Rödseth-Gupta theorem on binary partitions states that the following congruence holds: \(b(2^{s+2}n)-b(2^sn)\equiv 2^{\mu(s)}\pmod{2^{\mu(s)+1}}\), where \(\mu(s)=[(3s+4)/2]\) denotes the integer part of \((3s+4)/2\). In this paper the author presents a new proof of this fact, which seems to be technically easier than the previously known ones. His method allows to obtain some other properties and congruences of the sequence \(b(n)\). For instance, he proves that \(b(2^{s+4}n)+7b(2^{s+2}n)-8b(2^sn)\) is divisible by \([3s/2+8]\)th power of \(2\) if \(n\) is odd and \(s\geq 2\).