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 A367783 #47 Feb 17 2024 08:19:38 %S A367783 0,0,0,1,0,0,0,2,0,0,0,1,0,2,0,2,0,4,0,6,0,6,0,11,0,12,8,7,0,6,0,8,0, %T A367783 18,0,32,0,20,0,29,0,42,8,67,16,30,0,13,0,22,0,32,0,42,0,64,0,50,0,64 %N A367783 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 shifts. %C A367783 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 A367783 a(n) > 0 for even n >= 12. %C A367783 a(n) > 0 for odd n with natural density 1 (among odd numbers). %C A367783 For odd n, a(n) is divisible by 8. %H A367783 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 A367783 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 A367783 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 A367783 For n = 4 a(4) = 1 way to place 4 points is as follows: %e A367783 .xx. %e A367783 .xx. %e A367783 For n = 8 a(8) = 2 ways to place 8 points are as follows: %e A367783 ..x. %e A367783 .xxx %e A367783 xxx. %e A367783 .x.. %e A367783 (and its reflection with respect to a vertical axis). %e A367783 For n = 18 a(18) = 4 ways to place 18 points are as follows: %e A367783 ...x.. %e A367783 ..xxx. %e A367783 .xxxxx %e A367783 xxxxx. %e A367783 .xxx.. %e A367783 ..x... %e A367783 (and its reflection with respect to a vertical axis), and %e A367783 .....x.... %e A367783 ......x... %e A367783 .......x.. %e A367783 ....x...x. %e A367783 ...xxx...x %e A367783 x...xxx... %e A367783 .x...x.... %e A367783 ..x....... %e A367783 ...x...... %e A367783 ....x..... %e A367783 (and its reflection with respect to a vertical axis). %Y A367783 Cf. A000009, A000292, A005232, A369382. %K A367783 nonn,more %O A367783 1,8 %A A367783 _Giedrius Alkauskas_, Nov 30 2023 %E A367783 a(36) corrected by _Giedrius Alkauskas_, Feb 02 2024 %E A367783 a(49)-a(60) from _Giedrius Alkauskas_, Feb 06 2024