Modular Periodicity of Random Initialized Recurrences

Abstract

Classical studies of the Fibonacci sequence focus on its periodicity modulo $m$ (the Pisano periods) with canonical initialization. We investigate instead the complete periodic structure arising from all $m^2$ possible initializations in $(\mathbb{Z}/m\mathbb{Z})^2$. We discover perfect mirror symmetry between the Fibonacci recurrence $a_n = a_{n-1} + a_{n-2}$ and its parity transform $a_n = - a_{n-1} + a_{n-2}$ and observe fractal self-similarity in the extension from prime to prime power moduli. Additionally, we classify prime moduli based on their quadratic reciprocity and demonstrate that periodic sequences exhibit weight preservation under modular extension. Furthermore, we define a minima distribution $P(n)$ governed by Lucas ratios, which satisfies the symmetric relation $P(n)=P(1-n)$. For cyclotomic recurrences, we propose explicit counting functions for the number of distinct periods with connections to necklace enumeration. These findings imply potential connections to Viswanath's random recurrence, modular forms and L-functions.