A321851 -id:A321851 - cp's OEIS frontend

A258218 Number of length (4n-1) bit patterns with 2n ones that form circulant core of Hadamard matrices of size 4*n.

3, 14, 22, 30, 38, 46, 0, 248, 70, 0, 344, 94

Offset: 1

Ratko V. Tomic, May 23 2015

Numbers were obtained via brute force enumeration and checking of Hamming distances for all binomial(4*n-1,2*n) combinations of 4*n-1 length bit strings with exactly 2*n ones.
Each of a(n) bit patterns of length 4*n-1 when shifted 4*n-1 times forms rows of the (4*n-1) X (4*n-1) core of the normalized Hadamard matrix H(4*n).
The numbers a(n) are of the form k(n)*(4*n-1), where k(n) is 0, 1, or an even integer which varies with n. E.g., k=1 for H(4), k=2 for H(8) to H(24), k=0 for H(28) (i.e., no H(28) with circulant core exists), 8 for H(32), 2 for H(36), unknown even number >= 2 for H(40).
The sequence of 4*n numbers for nonzero values of a(n) (i.e., 4, 8, 12, 16, 20, 24, 32, 36, 248) appears to follow in order the subsets of sequences A034045, A010066 and A180490.
All a(n) patterns for n>1 are obtained from k(n)/2 seed patterns via 4*n-1 circular shifts of the seed pattern and their bit reversal.

			a(1)=3=1*(4*1-1), a(2)=14=2*(4*2-1), a(3)=22=2*(4*3-1), a(4)=30=2*15, a(7)=0, a(8)=248=8*31, a(9)=70=2*35, a(10)=0, a(11)=344=8*43, a(12)=94=2*47.

Minfeng Wang, C++ code for A258218
Wikipedia, Hadamard matrix

Cf. A034045, A010066 and A180490 for n values. A321851 concerns Hadamard matrices of quaternion form.

a(n) = k(n)*(4*n-1), where k(n) is an algorithmically defined function of n yielding 0, 1, or even integers. The algorithm for k(n) consists of enumeration of all combinations C(4*n-1,2*n) with counting of bit patterns that yield Hamming distances between the 2*n-1 circularly shifted pairs of exactly 2*n.

a(1)-a(9) confirmed and a(10)-a(12) extended by Minfeng Wang, Apr 25 2024

A319594 Number of solutions to dft(p)^2 + dft(q)^2 = (4n-3), where p and q are even sequences of length 2n-1, p(0)=0, p(k)=+1,-1 when k<>0, q(k) is +1,-1 for all k, and dft(x) denotes the discrete Fourier transform of x.

Original entry on oeis.org

2, 4, 4, 12, 12, 0, 12, 16, 0, 36, 24, 0, 20, 36, 0, 60

Offset: 1

Jeffery Kline, Dec 16 2018

Each solution (p,q) corresponds to a family of symmetric Hadamard matrices of size 8n-4.  To construct one member from this family, set A = circulant(p) + I, B = circulant(q), C = B, D = A - 2 I and H = [ [A, B, C, D], [B, D, -A, -C], [C, -A, -D, B], [D, -C, B, -A]]. Then A, B, C and D are symmetric and H is Hadamard and symmetric.
Since p and q are assumed to be even, dft(p) and dft(q) are real-valued.
2 divides a(n) for all n. If (p,q) is a solution, then (p,-q) is also a solution.
4 divides a(n) when n>1. If (p,q) is a solution, then (+/-p,+/-q) are also solutions. When n=1, p is the length-1 sequence, (0).
It is known that a(n)>0 for n=25, 26, 29.

			For n=1, the a(1)=2 solutions are ((0),(-)) and ((0),(+)). For n=2, the a(2)=4 solutions are ((0,-,-), (-,+,+)), ((0,-,-), (+,-,-)), ((0,+,+),(-,+,+)) and ((0,+,+), (+,-,-)).

Jeffery Kline, List of all pairs (p,q) that are counted by a(n), for 1<=n<=16.
Jeffery Kline, List of pairs (p,q) that establish a(n)>0, for n=25, 26, and 29.
Jeffery Kline, Geometric Search for Hadamard Matrices, Theoret. Comput. Sci. 778 (2019), 33-46.
J. Seberry and N.A. Balonin, The Propus Construction for Symmetric Hadamard Matrices, arXiv:1512.01732 [math.CO], 2015.
J. Seberry and N.A. Balonin, Two infinite families of symmetric Hadamard matrices, Australasian Journal of Combinatorics, 69 (2015), 349-357.

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

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.

Original entry on oeis.org

16, 96, 64, 256, 192, 1536, 960

Offset: 1

Jeffery Kline, Dec 18 2018

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.

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

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

A258218 Number of length (4n-1) bit patterns with 2n ones that form circulant core of Hadamard matrices of size 4*n.

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A319594 Number of solutions to dft(p)^2 + dft(q)^2 = (4n-3), where p and q are even sequences of length 2n-1, p(0)=0, p(k)=+1,-1 when k<>0, q(k) is +1,-1 for all k, and dft(x) denotes the discrete Fourier transform of x.

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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.

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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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cp's OEIS Frontend

A258218 Number of length (4*n-1) bit patterns with 2*n ones that form circulant core of Hadamard matrices of size 4*n.

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A319594 Number of solutions to dft(p)^2 + dft(q)^2 = (4n-3), where p and q are even sequences of length 2n-1, p(0)=0, p(k)=+1,-1 when k<>0, q(k) is +1,-1 for all k, and dft(x) denotes the discrete Fourier transform of x.

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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.

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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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A258218 Number of length (4n-1) bit patterns with 2n ones that form circulant core of Hadamard matrices of size 4*n.