On the Gap-sum and Gap-product Sequences of Integer Sequences

TL;DR

This paper investigates the properties of gap-sum and gap-product sequences derived from integer sequences, providing closed-form formulas and generating functions, with specific results for Horadam and elementary sequences.

Contribution

It introduces explicit formulas and generating functions for gap sequences, extending understanding of their structure and connections to known number sequences.

Findings

Closed-form expressions for gap-sum and gap-product sequences.

Generation of gap-sum sequences for Horadam sequences.

Identification of Fuss-Catalan-Raney numbers in elementary gap-product sequences.

Abstract

In this note, we explore two families of sequences associated to a suitable integer sequence: the gap-sum sequence and the gap-product sequence. These are the sums and the products of consecutive numbers not in the original sequence. We give closed forms for both, in terms of the original sequence, and in the case of Horadam sequences, we find the generating function of the gap-sum sequence. For some elementary sequences, we indicate that the gap-product sequences are given by the Fuss-Catalan-Raney numbers.