Around the Fibonacci Numeration System

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Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones. Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways. Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function. In addition, using a theorem of Fine and Wilf, we give a formula for R(n) in ... continued below

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Edson, Marcia Ruth May 2007.

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  • Edson, Marcia Ruth

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Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones. Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways. Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function. In addition, using a theorem of Fine and Wilf, we give a formula for R(n) in terms of binomial coefficients modulo two.

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  • May 2007

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  • Sept. 28, 2007, 9:55 p.m.

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  • Dec. 12, 2013, 1:49 p.m.

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Edson, Marcia Ruth. Around the Fibonacci Numeration System, dissertation, May 2007; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc3676/: accessed February 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .