The Hadamard product and recursively defined sequences

TL;DR

This paper presents methods for deriving explicit formulas for recursively defined sequences, including solutions for quadratic map-like recurrences and linear difference equations with variable coefficients, along with related integer sequences.

Contribution

It introduces nonrecurrent formulas for certain recursive sequences, notably solving quadratic map-like recurrences and linear difference equations with variable coefficients.

Findings

Explicit formula for quadratic map-like recurrence

Formulas for linear difference equations with variable coefficients

Analysis of integer sequences related to quadratic maps

Abstract

In this paper the approach to obtaining nonrecurrent formulas for some recursively defined sequences is illustrated. The most interesting result in the paper is the formula for the solution of quadratic map-like recurrence. Also, some formulas for the solutions of linear difference equations with variable coefficients are obtained. At the end of the paper, some integer sequences associated with a quadratic map are considered.