A185064 -id:A185064 - cp's OEIS frontend

A208929 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of equivalence classes of non-constructable Golay sequences of length L.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 1, 1, 44, 0

Offset: 1

N. J. A. Sloane, Mar 03 2012

Dragomir Z. Dokovic, Equivalence classes and representatives of Golay sequences, Discrete Math. 189 (1998), no. 1-3, 79-93. MR1637705 (99j:94031).

Cf. A185064, A208924-A208929.

A208924 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of Golay sequences of length L.

Original entry on oeis.org

4, 8, 32, 192, 128, 1536, 1088, 64, 15360, 9728, 512, 184320, 102912

Offset: 1

N. J. A. Sloane, Mar 03 2012

Dragomir Z. Dokovic, Equivalence classes and representatives of Golay sequences, Discrete Math. 189 (1998), no. 1-3, 79-93. MR1637705 (99j:94031).
Matthew G. Parker, Kenneth G. Paterson, and Chintha Tellambura, Golay Complementary Sequences, in Wiley Encyclopedia of Telecommunications, John G. Proakis, ed., Wiley, 2003; alternate link, January 19, 2004. See Table 1 p. 7.

Cf. A185064, A208925, A208926, A208927, A208928, A208929.

a(n) = A208925(n) + A208926(n).

a(11)-a(13) from Vincenzo Librandi, Nov 26 2020

A208927 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of equivalence classes of Golay sequences of length L.

Original entry on oeis.org

1, 1, 1, 5, 2, 36, 25, 1, 336, 220

Offset: 1

N. J. A. Sloane, Mar 03 2012

Dragomir Z. Dokovic, Equivalence classes and representatives of Golay sequences, Discrete Math. 189 (1998), no. 1-3, 79-93. MR1637705 (99j:94031).

Cf. A185064, A208924-A208929. A208927=A208928+A208929.

A208928 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of equivalence classes of constructable Golay sequences of length L.

Original entry on oeis.org

0, 0, 1, 5, 0, 34, 24, 0, 292, 220, 12, 3032, 2088, 128

Offset: 1

N. J. A. Sloane, Mar 03 2012

Dragomir Z. Dokovic, Equivalence classes and representatives of Golay sequences, Discrete Math. 189 (1998), no. 1-3, 79-93. MR1637705 (99j:94031).

Cf. A185064, A208924-A208929.

A208925 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of constructable Golay sequences of length L.

Original entry on oeis.org

0, 0, 32, 192, 0, 1408, 1024, 0, 12544, 9728, 512, 132608, 94720, 8192

Offset: 1

N. J. A. Sloane, Mar 03 2012

The definition sounds paradoxical: how can a(n) possibly be zero? The answer seems to be that a Golay sequence of length L can exist without being "constructable"! - N. J. A. Sloane, Nov 26 2020

Dragomir Z. Dokovic, Equivalence classes and representatives of Golay sequences, Discrete Math. 189 (1998), no. 1-3, 79-93. MR1637705 (99j:94031).

Cf. A185064, A208924, A208926, A208927, A208928, A208929.

A208926 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of non-constructable Golay sequences of length L.

Original entry on oeis.org

4, 8, 0, 0, 128, 128, 64, 64, 2816, 0, 0, 51712, 8192

Offset: 1

N. J. A. Sloane, Mar 03 2012

Dragomir Z. Dokovic, Equivalence classes and representatives of Golay sequences, Discrete Math. 189 (1998), no. 1-3, 79-93. MR1637705 (99j:94031).

Cf. A185064, A208924, A208925, A208927, A208928, A208929.

a(11)-a(13) from Vincenzo Librandi, Nov 26 2020

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.

Original entry on oeis.org

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.

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.

cp's OEIS Frontend

A208929 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of equivalence classes of non-constructable Golay sequences of length L.

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A208924 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of Golay sequences of length L.

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A208927 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of equivalence classes of Golay sequences of length L.

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A208928 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of equivalence classes of constructable Golay sequences of length L.

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A208925 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of constructable Golay sequences of length L.

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A208926 Let L = A185064(n) be the n-th length for which a Golay sequence exists; a(n) = number of non-constructable Golay sequences of length L.

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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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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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