The purpose of this article is to obtain some new infinite families of Toeplitz matrices, 7-matrices and generalized Pascal triangles whose leading principal minors form the Fibonacci, Lucas, Pell and Jacobsthal sequences. We also present a new proof for Theorem 3.1 in [R. Bacher. Determinants of matrices related to the Pascal triangle. J. Théor. Nombres Bordeaux, 14:19–41, 2002.].

It is the purpose of this article to present a triangular array of numbers similar to Pascal’s triangle and to prove a corresponding criterion for the twin prime pairs. A further goal is to place all this in the context of some classical orthogonal polynomials and to relate it to some recent work of John D’Angelo. To begin, and for the sake of completeness, we present a short proof of the Pasca...

The coefficients of the Pascal triangle were generalized in 1756 by de Moivre [5]. Each row of a Pascal triangle contains a sequence of numbers that are the coefficients of the power series expansion for the binary expression (l + x)^. The de Moivre formula [2], [4], [5], [6] derives the coefficients of the power series for the generalized expansion of (1 + x + x + • • • + x^"^). Thus, for inte...

À quoi reconnâıt-on qu’une suite est plus ou moins “compliquée” ? Une des traductions mathématiques de ce terme vague consiste à compter les facteurs ou blocs qui apparaissent dans cette suite, (voir par exemple [2]). Il y a naturellement bien d’autres approches possibles, qui dépendent en particulier à la fois des applications qu’on a à l’esprit ... et des quantités que l’on sait calculer ou e...

for 0 ≤ i, j ∈ N. The matrix P is hence the famous Pascal triangle yielding the binomial coefficients and can be recursively constructed by the rules p0,i = pi,0 = 1 for i ≥ 0 and pi,j = pi−1,j + pi,j−1 for 1 ≤ i, j. In this paper we are interested in (sequences of determinants of finite) matrices related to P . The present section deals with determinants of some minors of the above Pascal tria...

For any positive integer m let Fm = 2 2 + 1 be the mth Fermat number. In this short note we show that the only solutions of the diophantine equation Fm = ( n k ) are the trivial ones, i.e., those with k = 1 or n− 1.

for 0 ≤ i, j ∈ N. The matrix P is hence the famous Pascal triangle yielding the binomial coefficients and can be recursively constructed by the rules p0,i = pi,0 = 1 for i ≥ 0 and pi,j = pi−1,j + pi,j−1 for 1 ≤ i, j. In this paper we are interested in (sequences of determinants of finite) matrices related to P . The present section deals with some minors (determinants of submatrices) of the abo...

In this paper we introduce a new type of Pascal’s pyramids. The new object is called hyperbolic Pascal pyramid since the mathematical background goes back to the regular cube mosaic (cubic honeycomb) in the hyperbolic space. The definition of the hyperbolic Pascal pyramid is a natural generalization of the definition of hyperbolic Pascal triangle ([2]) and Pascal’s arithmetic pyramid. We descri...

We define the Pascal triangle of a discrete (gray scale) image as a pyramidal arrangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the e...

A binary triangle of size n is a triangle of zeroes and ones, with n rows, built with the same local rule as the standard Pascal triangle modulo 2. A binary triangle is said to be balanced if the absolute difference between the numbers of zeroes and ones that constitute this triangle is at most 1. In this paper, the existence of balanced binary triangles of size n, for all positive integers n, ...