A321338 -id:A321338 - cp's OEIS frontend

A321851 Number of solutions to |dft(a)^2 + dft(b)^2 + dft(d)^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.

16, 96, 480, 1600, 4800, 13824

Offset: 1

Jeffery Kline, Dec 19 2018

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 and D are circulant matrices formed by a, b and d, respectively, and C=fliplr(circulant(c)).
Each solution (a,b,c,d) also satisfies |dft(a)|^2 + |dft(b)|^2 + |dft(c)|^2 + |dft(d)^2| = 4n.
It is known that a(n) > 0 for 1 <= n <= 33 and n=35.
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(n) > A321338(n) when n > 2.

L. D. Baumert and M. Hall, Hadamard matrices of the Williamson type, Math. Comp. 19:91 (1965) 442-447.
Jeffery Kline, Geometric Search for Hadamard Matrices, Theoret. Comput. Sci. 778 (2019), 33-46.
Jeffery Kline, List of tuples (a,b,c,d) to demonstrate that a(n)>0, for 1<=n<=33 and n=35.

Cf. A020985, A185064, A319594, A321338.

Sequence A258218 concerns the Paley construction.

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.

Original entry on oeis.org

16, 96, 576, 1664, 4800, 23040

Offset: 1

Jeffery Kline, Dec 20 2018

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.

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.

Cf. A007299, A020985, A185064, A258218, A319594, A321338, A321851, A322639.

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.

Original entry on oeis.org

16, 96, 192, 896, 960, 4608, 6720

Offset: 1

Jeffery Kline, Dec 21 2018

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).

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.

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

cp's OEIS Frontend

A321851 Number of solutions to |dft(a)^2 + dft(b)^2 + dft(d)^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.

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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.

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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.

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