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arXiv · 2201.06636

On digital sequences associated with Pascal's triangle

Pierre Mathonet·Michel Rigo·Manon Stipulanti·Naïm Zénaïdi
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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.

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Pierre Mathonet, Michel Rigo, Manon Stipulanti, Naïm Zénaïdi. 2022-01-17. On digital sequences associated with Pascal's triangle. https://arxiv.org/abs/2201.06636

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