subject to the initial conditions A(l, 0) = 1, A(l, 1) ='2, with 4(n, /c) = 0 for & < 0 or k > n. This array has been called a Lucas triangle by Feinberg [1], because rising diagonals sum to give the Lucas numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ..., in contrast to the rising diagonals in the standard Pascal triangle where rising diagonals sum to give the Fibonacci numbers 1, 1, ...

1≤i,j≤n denote the n×n matrix formed by right justifying the first n rows of Pascal’s triangle. Let a denote the golden ratio (1+ √ 5)/2. We will show that the eigenvalues of R are λj = (−1)n+ja2j−n−1, 1 ≤ j ≤ n, (as conjectured in [1]), with corresponding eigenvectors uj = (uij)1≤i≤n where uij = ∑j k=1(−1) ( i−1 k−1 )(n−i j−k ) a2k−i−1. Since the eigenvalues are distinct, the eigenvectors are ...

In this paper, we look at a diophantine equation of the form un = ( x y ) , where (un)n≥0 is a binary recurrent sequence of integers. We show that if the pair of integers (x, y) belongs to a fixed line of the Pascal triangle, then the above equation has only finitely many positive integer solutions (n, x, y). A binary recurrent sequence of integers (un)n≥0 has u0, u1 ∈ Z and satisfies the recur...

Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schröder numbers and small Sch...

Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We als...

In a recent paper [1] it was shown a clean connection between the solution of the classical heat equation and the Tartaglia/Pascal/Yang-Hui triangle. When the time variable in the heat equation is substituted with the imaginary time, the heat equation becomes the Schrödinger equation of the quantum mechanics. So, a conjecture was put forward about a connection between the solution of the Schröd...