Representation of Natural Numbers as Sums of Generalised Fibonacci Numbers

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