Combinatorics on words; Pascal's triangle; Sierpinski’s triangle; Constructible regular polygon; Fermat numbers; Regular sequences; Automatic sequences; Base-p expansions; Pascal’s pyramid; Binomial coefficients; Evil and odious numbers; Nim sum
Résumé :
[en] 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.
Disciplines :
Mathématiques
Auteur, co-auteur :
Mathonet, Pierre ; Université de Liège - ULiège > Département de mathématique > Géométrie différentielle
Rigo, Michel ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Stipulanti, Manon ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Zenaïdi, Naïm ; Université de Liège - ULiège > Département de mathématique > Département de mathématique
Langue du document :
Anglais
Titre :
On digital sequences associated with Pascal's triangle
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