It is shown that the number of n × n n \times n integral triple diagonal matrices which are unimodular, positive definite and whose sub and super diagonal elements are all one, is the Catalan number ( n 2 n ) / ( n + 1 ) (_n^{2n})/(n + 1) . More generally, it is shown that if A is a fixed integral symmetric matrix and d is a fixed positive integer, then there are only finitely many integral diagonal matrices D such that A + D A + D is positive definite and det ( A + D ) = d \det (A + D) = d .