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 A322617 Number of solutions to |dft(a)^2 + dft(d)^2| + |dft(b)^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. 4
 16, 96, 576, 1664, 4800, 23040 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 and D are circulant matrices formed by a and d, respectively, and B=fliplr(circulant(b)) and C=fliplr(circulant(c)). The converse is not always true. To see this, set a=(-1, -1, -1, 1), b=(-1, -1, -1, 1), c=(-1, 1, 1, 1) and d=(1, -1, -1, -1). Then H is Hadamard but |dft(a)^2 + dft(d)^2| + |dft(b)^2 + dft(c)^2| = (16, 0, 16, 0). 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. LINKS Table of n, a(n) for n=1..6. L. D. Baumert and M. Hall, Hadamard matrices of the Williamson type, Math. Comp. 19:91 (1965) 442-447. W. H. Holzmann, H. Kharaghani and B. Tayfeh-Rezaie, Williamson matrices up to order 59, Des. Codes Cryptogr. 46 (2008), 343-352. Jeffery Kline, A complete list of solutions (a,b,c,d), for 1<=n<=5. Jeffery Kline, Geometric Search for Hadamard Matrices, Theoret. Comput. Sci. 778 (2019), 33-46. CROSSREFS Cf. A007299, A020985, A185064, A258218, A319594, A321338, A321851, A322639. Sequence in context: A091079 A321851 A185789 * A014352 A083295 A044267 Adjacent sequences: A322614 A322615 A322616 * A322618 A322619 A322620 KEYWORD nonn,more AUTHOR Jeffery Kline, Dec 20 2018 STATUS approved

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Last modified May 20 10:42 EDT 2024. Contains 372712 sequences. (Running on oeis4.)