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.

16, 96, 480, 1600, 4800, 13824

Offset: 1

Author

Jeffery Kline, Dec 19 2018

Keywords

• nonn
• more

Comments

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.

Links

• L. D. Baumert and M. Hall, Hadamard matrices of the Williamson type, Math. Comp. 19:91 (1965) 442-447.
• Jeffery Kline, Geometric Search for Hadamard Matrices, Theoret. Comput. Sci. 778 (2019), 33-46.
• Jeffery Kline, List of tuples (a,b,c,d) to demonstrate that a(n)>0, for 1<=n<=33 and n=35.

Crossrefs

Cf. A020985, A185064, A319594, A321338.

Sequence A258218 concerns the Paley construction.