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On the Number of Partitions of an Integer in the m -bonacci Base

Marcia EdsonLuca Q. Zamboni — 2006

Annales de l’institut Fourier

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...

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