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%I #53 Jan 30 2023 07:40:17
%S 16,96,64,256,192,1536,960
%N Number of solutions to dft(a)^2 + dft(b)^2 + dft(c)^2 + dft(d)^2 = 4n, where a,b,c,d are even +1,-1 sequences of length n and dft(x) denotes the discrete Fourier transform of x.
%C Each solution corresponds to a 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.
%C Since a,b,c and d are even, their discrete Fourier transforms are real-valued.
%C 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.
%C It appears that a(2n) > a(2n-1).
%C 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.
%H L. D. Baumert and M. Hall, <a href="https://doi.org/10.1090/S0025-5718-1965-0179093-2">Hadamard matrices of the Williamson type</a>, Math. Comp. 19:91 (1965) 442-447.
%H D. Z. Dokovic, <a href="https://doi.org/10.1016/0012-365X(93)90495-F">Williamson matrices of order 4n for n= 33, 35, 39</a>, Discrete mathematics (1993) May 15;115(1-3):267-71.
%H Jeffery Kline, <a href="/A321338/a321338_2.txt">A complete list of solutions (a,b,c,d)</a>, for 1<=n<=7.
%Y Cf. A007299, A020985, A185064, A258218, A319594, A321851, A322617, A322639.
%K nonn,more
%O 1,1
%A _Jeffery Kline_, Dec 18 2018
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