Shadow sequences of integers, from Fibonacci to Markov and back

TL;DR

This paper explores the concept of 'shadow sequences' that follow original integer sequences, such as Fibonacci and Markov sequences, revealing new relationships and patterns among these mathematical constructs.

Contribution

It introduces the novel idea of shadow sequences and demonstrates their existence for various well-known integer sequences, expanding understanding of their interconnections.

Findings

Shadow sequences exist for many classical integer sequences.

New relationships between Fibonacci, Markov, and other sequences are identified.

The concept opens avenues for further mathematical exploration of sequence interactions.

Abstract

The following general idea looks crazy. What if another integer sequence follows each integer sequence like a shadow? I will demonstrate that this is indeed the case, perhaps not for every integer sequence, but for many of them.