cp's OEIS Frontend

A368499 Number of non-congruent simple polygons with 2n sides on the unbounded chessboard such that each side is an edge of the corresponding knight graph.

%I A368499 #32 Mar 04 2024 04:28:42
%S A368499 3,13,178,3034,64877,1503790,36930111
%N A368499 Number of non-congruent simple polygons with 2n sides on the unbounded chessboard such that each side is an edge of the corresponding knight graph.
%C A368499 A knight graph is a graph that represents all legal moves of the knight chess piece on a chessboard. Each vertex of this graph represents a square of the chessboard, and each edge connects two squares that are a knight's move apart from each other.
%C A368499 Two polygons in the knight graph are called congruent if one can be transformed into the other by applying one or more of the operations of translation, rotation, and reflection on the chessboard; otherwise, they are non-congruent.
%C A368499 This sequence, in contrast to A366778, considers only simple, i.e., non-self-intersecting polygons.
%H A368499 Stoyan Kapralov, Valentin Bakoev, and Kaloyan Kapralov, <a href="https://doi.org/10.53656/math2023-2-1-alg">Algorithms for Construction and Enumeration of Closed Knight's Paths</a>, Mathematics and Informatics, 2, (2023), 107-114; <a href="https://arxiv.org/abs/2304.00565">arXiv:2304.00565 [math.CO]</a>, 2023.
%H A368499 Kaloyan Kapralov, <a href="/A368499/a368499.png">Example for n=3: the 13 non-congruent simple polygons</a>.
%e A368499 For n=2 the a(2)=3 solutions (in standard chess notation) are:
%e A368499   (a1,c2,d4,b3), (a2,c1,d2,c3), (a2,c1,d3,b3).
%e A368499 For n=3 the a(3)=13 solutions are:
%e A368499   (a1,b3,a5,c4,e3,c2), (a1,b3,a5,c6,b4,c2), (a1,b3,a5,c6,d4,c2),
%e A368499   (a1,b3,c5,e6,d4,c2), (a2,b4,c2,d4,e2,c1), (a2,b4,c6,d4,b3,c1),
%e A368499   (a2,b4,c6,d4,e2,c1), (a2,b4,c6,e5,d3,c1), (a2,b4,d5,c3,e2,c1),
%e A368499   (a2,b4,d5,f4,d3,c1), (a2,b4,d5,f4,e2,c1), (a2,c1,d3,f4,d5,c3),
%e A368499   (a2,c1,e2,g3,e4,c3).
%Y A368499 Cf. A366778, A001230, A234623, A254129, A356404.
%K A368499 nonn,hard,more
%O A368499 2,1
%A A368499 _Kaloyan Kapralov_, Dec 27 2023