
COMMENTS

Each solution (a,b,c,d) corresponds to a Hadamard matrix of quaternion type H = [[A, B, C, D], [B, A, D, C], [C, D, A, B], [D, C, B, A]], where A and D are circulant matrices formed by a and d, respectively, and B=fliplr(circulant(b)) and C=fliplr(circulant(c)). The converse is not always true. To see this, set a=(1, 1, 1, 1), b=(1, 1, 1, 1), c=(1, 1, 1, 1) and d=(1, 1, 1, 1). Then H is Hadamard but dft(a)^2 + dft(d)^2 + dft(b)^2 + dft(c)^2 = (16, 0, 16, 0).
16 is a divisor of a(n), for all n. If (a,b,c,d) is a solution, then each of the 16 tuples ((+)a, (+)b, (+)c, (+)d) is also a solution.
