Matrix Powers of Column-justified Pascal Triangles and Fibonacci Sequences

It is known that if Ln, respectively Rn9 are n x n matrices with the (/, j ) * entry the binomial coefficient (y~l)? respectively (^l)), then Ln = In (mod 2), respectively R„=In (mod 2), where In is the identity matrix of dimension n>\ (see, e.g., Problem PI 073 5 in the May 1999 issue of Arner. Math Monthly). The entries of Ln form a left-justified Pascal triangle and the entries of Rn result from taking the mirror-image of this triangle with respect to its first column. The questions we ask are: Can this result be extended to other primes or, better yet, is it possible to find a closed form for the entries of powers of Ln and Rnl Ln succumbs easily, as we shall see in our first result. Rn in turn fights back, since closed forms for its powers are not found. However, we show a beautiful connection between matrices similar to Rn and the Fibonacci numbers. If n 2, the connection is easily seen, since