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%I #33 Apr 28 2024 12:19:07
%S 3,14,22,30,38,46,0,248,70,0,344,94
%N Number of length (4*n-1) bit patterns with 2*n ones that form circulant core of Hadamard matrices of size 4*n.
%C 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.
%C 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).
%C 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).
%C 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.
%C 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.
%H Minfeng Wang, <a href="/A258218/a258218.cpp.txt">C++ code for A258218</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hadamard_matrix">Hadamard matrix</a>
%F 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.
%e 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.
%Y Cf. A034045, A010066 and A180490 for n values. A321851 concerns Hadamard matrices of quaternion form.
%K nonn,more
%O 1,1
%A _Ratko V. Tomic_, May 23 2015
%E a(1)-a(9) confirmed and a(10)-a(12) extended by _Minfeng Wang_, Apr 25 2024
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