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COMMENTS
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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, B and D are circulant matrices formed by a, b and d, respectively, and C=fliplr(circulant(c)).
Each solution (a,b,c,d) also satisfies |dft(a)|^2 + |dft(b)|^2 + |dft(c)|^2 + |dft(d)^2| = 4n.
It is known that a(n) > 0 for 1 <= n <= 33 and n=35.
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.
It appears that a(n) > A321338(n) when n > 2.
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