On digital sequences associated with Pascal's triangle

TL;DR

This paper investigates integer sequences derived from Pascal's triangle modulo a prime, revealing their structure, connections to known sequences, and extensions involving prime-related digit sums and Pascal's pyramid.

Contribution

It generalizes known relations of these sequences, shows their appearance as subsequences of 2-regular sequences, and explores their links to odious and evil numbers, Gray codes, and Pascal's pyramid.

Findings

Sequence is a subsequence of a 2-regular sequence

Connections to odious and evil numbers, Nim-sum, Gray codes

Extensions involving prime-related alternating digit sums

Abstract

We consider the sequence of integers whose $n$th term has base-$p$ expansion given by the $n$th row of Pascal's triangle modulo $p$ (where $p$ is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane's On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as a subsequence of a $2$-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involving alternating sum-of-digits modulo~$p$. This note ends with a discussion about Pascal's pyramid involving trinomial coefficients.