English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Zeckendorf's theorem states that every positive integer can be writtenuniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$,where we take $F_1=1$ and $F_2=2$. This has been generalized for any PositiveLinear Recurrence Sequence (PLRS), which informally is a sequence satisfying ahomogeneous linear recurrence with a positive leading coefficient andnon-negative integer coefficients. These decompositions are generalizations ofbase $B$ decompositions. In this and the follow-up paper, we provide twoapproaches to investigate linear recurrences with leading coefficient zero,followed by non-negative integer coefficients, with differences between indicesrelatively prime (abbreviated ZLRR). The first approach involves generalizingthe definition of a legal decomposition for a PLRS found in Kolo\u{g}lu, Kopp,Miller and Wang. We prove that every positive integer $N$ has a legaldecomposition for any ZLRR using the greedy algorithm. We also show that aspecific family of ZLRRs loses uniqueness of decompositions. The secondapproach converts a ZLRR to a PLRR that has the same growth rate. We developthe Zeroing Algorithm, a powerful helper tool for analyzing the behavior oflinear recurrence sequences. We use it to prove a very general result thatguarantees the possibility of conversion between certain recurrences, anddevelop a method to quickly determine whether certain sequences diverge to$+\infty$ or $-\infty$, given any real initial values. This paper investigatesthe first approach.

Generalizing Zeckendorf's Theorem to Homogeneous Linear Recurrences, I