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A322617 Number of solutions to |dft(a)^2 + dft(d)^2| + |dft(b)^2 + dft(c)^2| = 4n, where a,b,c,d are +1,-1 sequences of length n and dft(x) denotes the discrete Fourier transform of x.

%I A322617 #25 Jan 30 2023 07:40:26
%S A322617 16,96,576,1664,4800,23040
%N A322617 Number of solutions to |dft(a)^2 + dft(d)^2| + |dft(b)^2 + dft(c)^2| = 4n, where a,b,c,d are +1,-1 sequences of length n and dft(x) denotes the discrete Fourier transform of x.
%C A322617 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).
%C A322617 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.
%H A322617 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 A322617 W. H. Holzmann, H. Kharaghani and B. Tayfeh-Rezaie, <a href="http://math.ipm.ac.ir/~tayfeh-r/papersandpreprints/Williamson.pdf">Williamson matrices up to order 59</a>, Des. Codes Cryptogr. 46 (2008), 343-352.
%H A322617 Jeffery Kline, <a href="/A322617/a322617_1.txt">A complete list of solutions (a,b,c,d)</a>, for 1<=n<=5.
%H A322617 Jeffery Kline, <a href="https://doi.org/10.1016/j.tcs.2019.01.025">Geometric Search for Hadamard Matrices</a>, Theoret. Comput. Sci. 778 (2019), 33-46.
%Y A322617 Cf. A007299, A020985, A185064, A258218, A319594, A321338, A321851, A322639.
%K A322617 nonn,more
%O A322617 1,1
%A A322617 _Jeffery Kline_, Dec 20 2018