This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382693 #47 May 28 2025 01:04:01
%S A382693 1,2,4,20,234,6308,374586,47740076,12788143462,7090729971308
%N A382693 Number of dense symmetric binary relations on {1,...,n}.
%C A382693 A binary relation R is usually called dense if for all R-related x and y there exists z such that x and z and also z and y are R-related (see Wikipedia link), but here we define a symmetric binary relation R on {1,...,n} to be "dense" (see the second page of the paper in the links) iff for all x and y we have (zRx implies zRy for all z) implies (x <= y).
%C A382693 Equivalently, as mentioned in a SeqFan post (see link), for all x < y, some z satisfies (not zRx) AND zRy.
%C A382693 Note that terms were added for n = 7 and 8 in a SeqFan post (see link).
%H A382693 Francesco Ciraulo and Giovanni Sambin, <a href="https://doi.org/10.1016/j.jpaa.2010.02.002">The overlap algebra of regular opens</a>, Journal of Pure and Applied Algebra, Volume 214, Issue 11, November 2010, Pages 1988-1995.
%H A382693 Seqfan, <a href="https://groups.google.com/g/seqfan/c/58wBM59Om_8/m/YwoIkrrYAgAJ">number of dense symmetric binary relations on {1,...,n} (A382693)</a>.
%H A382693 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dense_order#Generalizations">Dense Order, Generalizations</a>.
%Y A382693 Cf. A382839 (with symmetry added).
%K A382693 nonn,more
%O A382693 0,2
%A A382693 _Mark Bowron_, Apr 03 2025
%E A382693 a(9) from _Christian Sievers_, May 26 2025