OFFSET
1,2
COMMENTS
Former name: Number of n X n symmetric binary matrices with zero diagonal and no three-node loops x(i,j)*x(j,k)*x(k,i) = 1, i < j < k.
From Brendan McKay, Jun 11 2021: (Start)
EXP transform of A345218.
Labeled version of A006785. (End)
a(n) is the number of sign mappings X:([n] choose 2) -> {+,-} such that for any ordered 3-tuple a<b<c we have X(ab)X(ac)X(bc) not equal to +++. - Manfred Scheucher, Jan 05 2024
LINKS
Tobias Boege and Thomas Kahle, Construction Methods for Gaussoids, arXiv:1902.11260 [math.CO], 2019.
Simon Dreyer, Antoine Genitrini, and Mehdi Naima, Asymptotic Enumeration of Labeled Triangle-Free Graphs through the Combinatorics of Directed Acyclic Graphs of Shortest Paths, hal:05609965, 2026.
P. Erdős, D. J. Kleitman, and B. L. Rothschild, Asymptotic enumeration of k_n-free graphs. In Colloquio Internazionale sulle Teorie Combinatorie, (Rome, 1973), Tomo II, Atti dei Convegni Lincei, No. 17, pp. 19-27. Accad. Naz. Lincei, Rome.
Falk Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 8c665c7.
Vaclav Kotesovec, Plot of a(n) / (2^(n^2/4+n-1/2)/sqrt(Pi*n)) for n = 1..16
FORMULA
a(n) ~ c * 2^(n^2/4+n-1/2)/sqrt(Pi*n), where c = Sum_{k = -oo..oo} 2^(-k^2) = EllipticTheta[3, 0, 1/2] = 2.128936827211877... if n is even and c = Sum_{k = -oo..oo} 2^(-(k+1/2)^2) = EllipticTheta[2, 0, 1/2] = 2.12893125051302... if n is odd. - Mehdi Naima, Jun 05 2026 [Editor note: Current data do not permit a numerical reconstruction of the asymptotic formula at this time (Vaclav Kotesovec). However, for small values, the number of triangle-free graphs that are not bipartite may differ substantially from those for larger values (Mehdi Naima).]
EXAMPLE
Some solutions for n=4:
0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0
1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1
0 1 0 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0
0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0
CROSSREFS
KEYWORD
nonn,more,changed
AUTHOR
R. H. Hardin, Jun 11 2012
EXTENSIONS
a(11)-a(13) added using tinygraph by Falk Hüffner, Jun 19 2018
a(14)-a(15) added using tinygraph by Falk Hüffner, Oct 28 2019
a(16) added by Brendan McKay, Sep 15 2020
Name changed to the one suggested by Falk Hüffner and Brendan McKay, Jun 11 2021
STATUS
approved