An Algorithm for Calculating Terms of a Number Sequence using an Auxiliary Sequence

TL;DR

This paper presents an algorithm for computing terms of linear recursive sequences using an auxiliary sequence, enabling calculation without closed-form expressions, and employs generating functions for analysis.

Contribution

It introduces a novel algorithm for calculating sequence terms without closed-form solutions, expanding computational methods for linear recursions.

Findings

Algorithm enables term calculation without closed-form expressions

Uses generating functions to analyze sequences

Applicable to sequences with arbitrary index differences

Abstract

Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is derived for calculating the nth term even without an expression in closed form.