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A321338 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. 4
16, 96, 64, 256, 192, 1536, 960 (list; graph; refs; listen; history; text; internal format)
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.
Since a,b,c and d are even, their discrete Fourier transforms are real-valued.
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(2n) > a(2n-1).
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.
L. D. Baumert and M. Hall, Hadamard matrices of the Williamson type, Math. Comp. 19:91 (1965) 442-447.
D. Z. Dokovic, Williamson matrices of order 4n for n= 33, 35, 39, Discrete mathematics (1993) May 15;115(1-3):267-71.
Jeffery Kline, A complete list of solutions (a,b,c,d), for 1<=n<=7.
Sequence in context: A333511 A320406 A241936 * A128702 A322639 A277044
Jeffery Kline, Dec 18 2018

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