A258218 Number of length (4*n-1) bit patterns with 2*n 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

Author

Ratko V. Tomic, May 23 2015

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

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

Crossrefs

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

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.

Extensions

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