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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.

%I A319594 #49 Jan 30 2023 04:45:59
%S A319594 2,4,4,12,12,0,12,16,0,36,24,0,20,36,0,60
%N 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.
%C A319594 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.
%C A319594 Since p and q are assumed to be even, dft(p) and dft(q) are real-valued.
%C A319594 2 divides a(n) for all n. If (p,q) is a solution, then (p,-q) is also a solution.
%C A319594 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).
%C A319594 It is known that a(n)>0 for n=25, 26, 29.
%H A319594 Jeffery Kline, <a href="/A319594/a319594_1.txt">List of all pairs (p,q)</a> that are counted by a(n), for 1<=n<=16.
%H A319594 Jeffery Kline, <a href="/A319594/a319594_2.txt">List of pairs (p,q)</a> that establish a(n)>0, for n=25, 26, and 29.
%H A319594 Jeffery Kline, <a href="https://doi.org/10.1016/j.tcs.2019.01.025">Geometric Search for Hadamard Matrices</a>, Theoret. Comput. Sci. 778 (2019), 33-46.
%H A319594 J. Seberry and N.A. Balonin, <a href="https://arxiv.org/abs/1512.01732">The Propus Construction for Symmetric Hadamard Matrices</a>, arXiv:1512.01732 [math.CO], 2015.
%H A319594 J. Seberry and N.A. Balonin, <a href="https://ajc.maths.uq.edu.au/pdf/69/ajc_v69_p349.pdf">Two infinite families of symmetric Hadamard matrices</a>, Australasian Journal of Combinatorics, 69 (2015), 349-357.
%e A319594 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,+,+), (+,-,-)).
%Y A319594 Cf. A007299, A020985, A185064, A258218, A321851, A322617, A322639.
%K A319594 nonn,more
%O A319594 1,1
%A A319594 _Jeffery Kline_, Dec 16 2018