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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)
 OFFSET 1,1 COMMENTS 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. LINKS 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. CROSSREFS 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 KEYWORD nonn,more AUTHOR Jeffery Kline, Dec 18 2018 STATUS approved

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