The Eigenvectors of the Right-Justified Pascal Triangle

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 linearly independent and so form an invertible matrix that diagonalizes R. Scaling the eigenvectors to vj = (−1)jan−j/(1 + a)uj yields a diagonalizing matrix V (V −1RV = diag(λj) n j=1) with a remarkable property: V −1 = V . This makes it easy to write down explicit three-summation formulas for the entries of powers of R.