

A140426


Number of multisymmetric 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
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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 uppertriangular coefficients satisfy a_{i,j}=a_{i1,j1}+a_{i1,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 doublysymmetric 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 <= n1} is a multisymmetric matrix, that is a doublysymmetric matrix where each row of its uppertriangular part is a symmetric sequence.
We prove that the multisymmetric 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], 20082009.
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((n3)/6) for n odd.
G.f.: ( 1x2*x^2x^32*x^42*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



