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.
翻译:我们考虑的是以美元计算第一期的整数序列,其基价-p美元增长幅度由Pascal三角形元元(美元是质数)第1行的美元表示。我们首先提出并概括了有关该序列的众所周知的关系。然后,在Sloane的Integer序列的Line百科全书的大力帮助下,我们发现它自然地是一个2美元经常序列的子序列。它的研究提供了有趣的关系,令人惊讶地涉及恶恶恶的数字、Nim-sum甚至灰色代码。此外,我们研究了从质数中产生的类似序列,涉及交替数数的数之和Mmodulo~p$。本说明最后讨论了Pascal的金字塔,涉及三角系数。