A Generalization Of Fibonacci Far-Difference Representations And Gaussian Behavior

A natural generalization of base B expansions is Zeckendorf's Theorem, which states that every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers {F-n}, with Fn+1 = F-n + Fn-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 +/- 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.In this paper 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-n+1((k)) = S-n((k)) + S-n-k((k)) for k >= 0. Every integer can be written uniquely as a sum of the +/- S-n((k))'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. Let I-n = (R-k(n - 1), R-k(n)] with R-k(l) = Sigma S-0 infinity, with a computable correlation coefficient. We next explore the distribution of gaps between summands, and show that for any k the probability of finding a gap of length j >= 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 (see Definition 1.11) for any positive integers s, d.