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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 an 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.


Table of n, a(n) for n=1..7.

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.


Cf. A007299, A020985, A185064, A258218, A319594, A321851, A322617, A322639.

Sequence in context: A318021 A320406 A241936 * A128702 A322639 A277044

Adjacent sequences:  A321335 A321336 A321337 * A321339 A321340 A321341




Jeffery Kline, Dec 18 2018



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Last modified October 16 01:30 EDT 2019. Contains 328038 sequences. (Running on oeis4.)