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Representation of Natural Numbers as Sums of Generalised Fibonacci Numbers
Author(s) -
Daykin D. E.
Publication year - 1960
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s1-35.2.143
Subject(s) - fibonacci number , natural number , citation , representation (politics) , computer science , natural (archaeology) , combinatorics , arithmetic , information retrieval , mathematics , discrete mathematics , library science , history , archaeology , politics , political science , law
The well-known observation of Zeckendorf is that every positive integer N has a unique representation N = u. +u. + • • • +u . , where (1) ij ^ 1 and i i ^ 2 for i 4= v < d , and ju {• is the Fibonacci sequence ••• , 0 , 0 , 1 , 2 , 3 , 5 , 8 , 1 3 , " defined by u = 0 for n 4 0 , n (2) { ui = 1 , u2 = 2, and u , = u + u ., for n =̂ 2 . n+1 n n-1 Existence of such a representation follows from (2), and its uniqueness follows easily from the identity