A Generalization of Fibonacci Far-Difference Representations and Gaussian Behavior

Abstract

A natural generalization of base B expansions is Zeckendorf's Theorem: every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, with $F_{n+1} = F_n + F_{n-1}$ and $F_1=1, F_2=2$. If instead we allow the coefficients of the Fibonacci numbers in the decomposition to be zero or $\pm 1$, the resulting expression is known as the far-difference representation. Alpert proved that a far-difference representation exists and is unique under certain restraints that generalize non-consecutiveness, specifically that two adjacent summands of the same sign must be at least 4 indices apart and those of opposite signs must be at least 3 indices apart. We prove that a far-difference representation can be created using sets of Skipponacci numbers, which are generated by recurrence relations of the form $S^{(k)}_{n+1} = S^{(k)}_{n} + S^{(k)}_{n-k}$ for $k \ge 0$. Every integer can be written uniquely as a sum of the $\pm S^{(k)}_n $'s such that every two terms of the same sign differ in index by at least 2k+2, and every two terms of opposite signs differ in index by at least k+2. Additionally, we prove that the number of positive and negative terms in given Skipponacci decompositions converges to a Gaussian, with a computable correlation coefficient that is a rational function of the smallest root of the characteristic polynomial of the recurrence. The proof uses recursion to obtain the generating function for having a fixed number of summands, which we prove converges to the generating function of a Gaussian. We next explore the distribution of gaps between summands, and show that for any k the probability of finding a gap of length $j \ge 2k+2$ decays geometrically, with decay ratio equal to the largest root of the given k-Skipponacci recurrence. We conclude by finding sequences that have an (s,d) far-difference representation for any positive integers s,d.