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A258218
Number of length (4*n-1) bit patterns with 2*n ones that form circulant core of Hadamard matrices of size 4*n.
6
3, 14, 22, 30, 38, 46, 0, 248, 70, 0, 344, 94
OFFSET
1,1
COMMENTS
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.
FORMULA
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.
EXAMPLE
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.
CROSSREFS
Cf. A034045, A010066 and A180490 for n values. A321851 concerns Hadamard matrices of quaternion form.
Sequence in context: A071836 A361909 A300216 * A255219 A226341 A024473
KEYWORD
nonn,more
AUTHOR
Ratko V. Tomic, May 23 2015
EXTENSIONS
a(1)-a(9) confirmed and a(10)-a(12) extended by Minfeng Wang, Apr 25 2024
STATUS
approved