
COMMENTS

Each solution corresponds to an Hadamard matrix of quaternion type. That is, if H = [[A, B, C, D], [B, A, D, C], [C, D, A, B], [D, C, B, A]], where A,B,C, and D are circulant matrices formed from a,b,c and d, respectively, then H is Hadamard.
Since a,b,c and d are even, their discrete Fourier transforms are realvalued.
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(2n) > a(2n1).
A321851(n) >= a(n), A322617(n) >= a(n) and A322639(n) >= a(n). Every solution that is counted by a(n) is also counted by A321851(n), A322617(n) and A322639(n), respectively.


LINKS

Table of n, a(n) for n=1..7.
L. D. Baumert and M. Hall, Hadamard matrices of the Williamson type, Math. Comp. 19:91 (1965) 442447.
D. Z. Dokovic, Williamson matrices of order 4n for n= 33, 35, 39 Discrete mathematics. (1993) May 15;115(13):26771.
Jeffery Kline, A complete list of solutions (a,b,c,d), for 1<=n<=7.
