On the Number of Partitions of an Integer in the m -bonacci Base
Marcia Edson, Luca Quardo Zamboni · Annales de l’institut Fourier · 2006
For each m ≥ 2 , we consider the m -bonacci numbers defined by F k = 2 k for 0 ≤ k ≤ m - 1 and F k = F k - 1 + F k - 2 + ⋯ + F k - m for k ≥ 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 ( n ) 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 ( n ) involving sums of binomial coefficients modulo 2 . In addition we show that this formula may be used to determine the number of partitions of n in more general numeration systems including generalized Ostrowski number systems in connection with Episturmian words.