Notes on Fibonacci Partitions

Abstract. Let f1 = 1, f2 = 2, f3 = 3, f4 = 5, . . . be the sequence of Fibonacci numbers. It is well known that for any natural n there is a unique expression n = fi1 + fi2 + · · · + fiq , such that ia+1 − ia > 2 for a = 1, 2, . . . , q− 1 (Zeckendorf Theorem). By means of it we find an explicit formula for the quantity Fh(n) of partitions of n with h summands, all parts of them are the pairwise different Fibonacci numbers. This formula is used for an investigation of the functions F (n) = ∑ ∞ h=1 Fh(n) and χ(n) = ∑ ∞ h=1 (−1)Fh(n). They are interpreted by means of representation of rational numbers as some continues fractions. Using this approach we define a canonical action of monoid Z2 × Z2 × N (see text for the notations) on the set of natural numbers, the set orbits of that is also a monoid, freely generated by the set Q/Z, and such that F (n) is invariant under this action. A fundamental domain of the action is found, and the following results are established: the formula χ(n) = 0,±1, a theorem on ”Fibonacci random distribution” of n with F (n) = k, the estimation F (n) 6 √ n+ 1, and it is shown that limN→∞(χ (1) + · · ·+χ2(N))/N = 0. In addition, an algorithm to find a minimal n with F (n) = k is provided.