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For each $m\ge 2,$ we consider the $m$-bonacci numbers defined by $F_k=2^k$ for $0\le k\le m-1$ and $F_k=F_{k-1}+F_{k-2}+\cdots +F_{k-m}$ for $k\ge m.$ When $m=2,$ these are the usual Fibonacci numbers. Every positive integer $n$ may be expressed as a sum of distinct $m$-bonacci numbers in one or more different ways. Let $R_m\left(n\right)$ be the number of partitions of $n$ as a sum of distinct $m$-bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for $R_m\left(n\right)$ involving sums of binomial coefficients modulo $2.$ In addition we show that this formula may be used to determine the number of partitions...