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A140426 Number of multi-symmetric Steinhaus matrices of size n. 0
1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 4, 4, 4, 8, 4, 8, 8, 8, 8, 16, 8, 16, 16, 16, 16, 32, 16, 32, 32, 32, 32, 64, 32, 64, 64, 64, 64, 128, 64, 128, 128, 128, 128, 256, 128, 256, 256, 256, 256, 512, 256, 512, 512, 512, 512, 1024, 512, 1024, 1024, 1024, 1024, 2048, 1024, 2048, 2048, 2048, 2048, 4096, 2048 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Theorem 3.7, p. 9, of Chappelon.

Abstract: A Steinhaus matrix is a binary square matrix of size n which is symmetric, with diagonal of zeros and whose upper-triangular coefficients satisfy a_{i,j}=a_{i-1,j-1}+a_{i-1,j} for all 2 <= i<j <= n. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix.

We give a short new proof of a theorem, due to Dymacek, which states that even Steinhaus graphs, i.e., those with all vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek conjectured that the complete graph on two vertices K_2 is the only regular Steinhaus graph of odd degree.

Using Dymacek's theorem, we prove that if (a_{i,j})_{1 <= i,j <= n} is a Steinhaus matrix associated to a regular Steinhaus graph of odd degree then its submatrix (a_{i,j})_{2 <= i,j <= n-1} is a multi-symmetric matrix, that is a doubly-symmetric matrix where each row of its upper-triangular part is a symmetric sequence.

We prove that the multi-symmetric Steinhaus matrices of size $n$ whose Steinhaus graphs are regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend on ceil (n/24} parameters for every even number n and on ceil (n/30} parameters in the odd case. This result permits us to verify the Dymacek's conjecture up to 1500 vertices in the odd case.

LINKS

Table of n, a(n) for n=0..69.

Jonathan Chappelon, Regular Steinhaus graphs of odd degree, arXiv:0806.2779 [math.CO], 2008-2009.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2)

FORMULA

a(n) = 2^ceiling(n/6) for n even, 2^ceiling((n-3)/6) for n odd.

G.f.: ( -1-x-2*x^2-x^3-2*x^4-2*x^5 ) / ( -1+2*x^6 ). - R. J. Mathar, Jan 22 2011

CROSSREFS

Sequence in context: A062610 A025801 A060548 * A146879 A231577 A277210

Adjacent sequences:  A140423 A140424 A140425 * A140427 A140428 A140429

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Jun 18 2008

STATUS

approved

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Last modified February 21 04:22 EST 2018. Contains 299389 sequences. (Running on oeis4.)