On digital sequences associated with Pascal's triangle

01/17/2022

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by   Pierre Mathonet, et al.

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We consider the sequence of integers whose nth term has base-p expansion given by the nth 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.