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 A382839 #23 May 28 2025 01:04:08
%S A382839 1,2,7,114,9602,3962940,7516789560,62622777447552,2221417812173570640
%N A382839 Number of dense binary relations on {1,...,n}.
%C A382839 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 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 A382839 Equivalently, as mentioned in a SeqFan post (see link), for all x < y, some z satisfies (not zRx) AND zRy.
%C A382839 Brendan McKay added terms for n = 6 and 7 in a SeqFan post (see link).
%H A382839 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 A382839 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> and <a href="https://groups.google.com/g/seqfan/c/58wBM59Om_8/m/mcUFKo5vAwAJ">also</a>.
%H A382839 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dense_order#Generalizations">Dense Order, Generalizations</a>.
%Y A382839 Cf. A382693 (without symmetry).
%K A382839 nonn,more
%O A382839 0,2
%A A382839 _Mark Bowron_, Apr 06 2025
%E A382839 a(8) from _Christian Sievers_, May 26 2025