A322639 Number of solutions to |dft(a)^2 + dft(b)^2 + dft(c)^2 + dft(d)^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, 192, 896, 960, 4608, 6720

Offset: 1

Author

Jeffery Kline, Dec 21 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, C and D are circulant matrices formed by a, b, c and d, respectively.

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.

a(n) >= A321338(n). Every solution (a,b,c,d) that is counted by A321338(n) is also counted by a(n).

Links

• Jeffery Kline, A complete list of solutions (a,b,c,d), for 1<=n<=7.
• Jeffery Kline, List of tuples (a,b,c,d) to demonstrate that a(n)>0, for 1<=n<=22 and n=24.

Crossrefs

Cf. A007299, A020985, A185064, A258218, A319594, A321338, A321851, A322617.