A321851 - OEIS

A321851

Number of solutions to |dft(a)^2 + dft(b)^2 + dft(d)^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 #62 Jan 30 2023 07:40:22

%S 16,96,480,1600,4800,13824

%N Number of solutions to |dft(a)^2 + dft(b)^2 + dft(d)^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 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)).

%C Each solution (a,b,c,d) also satisfies |dft(a)|^2 + |dft(b)|^2 + |dft(c)|^2 + |dft(d)^2| = 4n.

%C It is known that a(n) > 0 for 1 <= n <= 33 and n=35.

%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(n) > A321338(n) when n > 2.

%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 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.

%H Jeffery Kline, <a href="/A321851/a321851_1.txt">List of tuples (a,b,c,d) to demonstrate that a(n)>0</a>, for 1<=n<=33 and n=35.

%Y Cf. A020985, A185064, A319594, A321338.

%Y Sequence A258218 concerns the Paley construction.

%K nonn,more

%O 1,1

%A _Jeffery Kline_, Dec 19 2018