Powers of two in generalized fibonacci sequences

The $k-$generalized Fibonacci sequence $\big(F_{n}^{(k)}\big)_{n}$ resembles the Fibonacci sequence in that it starts with $0,\ldots,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we are interested in finding powers of two that appear in $k-$generaliz...