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 A369382 #35 Mar 27 2024 20:12:45 %S A369382 0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,2,0,2,0,2,0,4,0,4,1,3,0,2,0,3,0,6, %T A369382 0,10,0,6,0,9,0,12,1,18,2,9,0,5,0,7,0,8,0,12,0,18,0,14,0,17 %N A369382 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to lattice symmetry. %C A369382 A monotone path is a lattice path consisting of east and north unit steps or a path consisting of east and south unit steps. When counting, points lying on the path itself are discarded. %C A369382 Related to A367783, only sets obtained by rotation and reflection are considered to be the same. %C A369382 For odd n, a(n) = A367783(n)/8. %C A369382 For even n, 8 * a(n) >= A367783(n). %C A369382 a(n) > 0 for even n >= 12. %C A369382 a(n) > 0 for odd n with natural density 1 (among odd numbers). %H A369382 Giedrius Alkauskas, <a href="https://arxiv.org/abs/2302.01137">Friendly paths for finite subsets of plane integer lattice. I</a>, arXiv:2302.01137 [math.CO], 2024. %H A369382 Giedrius Alkauskas, <a href="https://www.jstor.org/stable/10.4169/000298910x476103">Problem 11484</a>, Problems and solutions, Amer. Math. Monthly, 117 (2) February (2010), p. 182. %H A369382 Giedrius Alkauskas, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.119.02.161?seq=8">Friendly paths. Problem 11484</a>, Problems and solutions, Amer. Math. Monthly, 119 (2) February (2012), 167-168. %e A369382 For n = 4, a(4) = 1 way to place 4 points is as follows: %e A369382 .xx. %e A369382 .xx. %e A369382 For n = 14, a(14) = 1 way to place 14 points is as follows: %e A369382 ...x.. %e A369382 ..x.x. %e A369382 .xxx.x %e A369382 x.xxx. %e A369382 .x.x.. %e A369382 ..x... %e A369382 For n = 27, a(27) = 1 way to place 27 points is as follows: %e A369382 ....x.... %e A369382 ...x..... %e A369382 ..x...... %e A369382 .x..xx... %e A369382 x..xxxx.. %e A369382 ..xxxxxxx %e A369382 ...xxxxx. %e A369382 ....xxx.. %e A369382 .....x... %Y A369382 Cf. A000009, A000292, A005232, A367783. %K A369382 nonn,more %O A369382 1,18 %A A369382 _Giedrius Alkauskas_, Jan 22 2024