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 A165143 #26 Jan 14 2025 09:08:08 %S A165143 8,10,16,20,22,32,40,48 %N A165143 Length of longest cyclic knight path on an n X n chessboard that is determined (up to starting point and direction) by the fields visited. %C A165143 More mathematically, a(n) is also the biggest possible number of vertices (or edges) of a circular graph in the plane with all vertices being distinct and having integer coordinates between 1 and n, inclusive, and all edges having length sqrt(5). %C A165143 Opinions on a(1) and a(2) may differ: The only possible circular path consists of staying at the starting field, hence we have 1 field visited (thus a(1) = a(2) = 1) or it takes 0 knight moves to complete the tour (thus a(1) = a(2) = 0). The graph with one vertex and no edges is not considered a cyclic graph since all vertices must have degree two; even adding a loop edge would produce an edge of length 0 (thus a(1) and a(2) are either undefined or = 0 if one accepts the empty graph as cyclic). %C A165143 Put another way, a(n) is the length of a longest induced cycle in the n X n knight graph. - _Pontus von Brömssen_, Oct 02 2022 %D A165143 Thomas Dawson, Échecs Féeriques, L'Échiquier, volume 2, issue 2, 1930; issue 3, 1931. %D A165143 Donald E. Knuth, The Art of Computer Programming, Volume 4B, Combinatorial Algorithms, Part 2, Addison-Wesley, 2023. See exercise 7.2.2.1-172 and its solution. %H A165143 Nikolai Beluhov, <a href="https://arxiv.org/abs/2301.01152">Snake paths in king and knight graphs</a>, arXiv:2301.01152 [math.CO], 2023. %H A165143 Sequence mentioned in the related <a href="https://problemcorner.missouristate.edu/Sol12_0708.html">Solution to problem #12</a>, Missoury State Math Dept. Challenge Problem Archive, 2007/08. %F A165143 From _Pontus von Brömssen_, Jan 30 2023: (Start) %F A165143 a(n) = n^2/2 + O(n) (Beluhov 2023). %F A165143 a(n) <= A357500(n)+1. %F A165143 (End) %e A165143 The eight non-center fields of a 3 X 3 chessboard can be visited cyclically in a unique manner up to selection of starting point and clockwise vs. counterclockwise direction; since the center field cannot be part of a (longer) cycle, we have a(3) = 8. %Y A165143 Cf. A297666, A357357, A357500, A357501. %K A165143 more,nonn %O A165143 3,1 %A A165143 _Hagen von Eitzen_, Sep 05 2009