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.

3, 13, 178, 3034, 64877, 1503790, 36930111

Offset

2,1

Comments

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.

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.

This sequence, in contrast to A366778, considers only simple, i.e., non-self-intersecting polygons.

Links

Table of n, a(n) for n=2..8.

Stoyan Kapralov, Valentin Bakoev, and Kaloyan Kapralov, Algorithms for Construction and Enumeration of Closed Knight's Paths, Mathematics and Informatics, 2, (2023), 107-114; arXiv:2304.00565 [math.CO], 2023.

Kaloyan Kapralov, Example for n=3: the 13 non-congruent simple polygons.

Example

For n=2 the a(2)=3 solutions (in standard chess notation) are:
  (a1,c2,d4,b3), (a2,c1,d2,c3), (a2,c1,d3,b3).
For n=3 the a(3)=13 solutions are:
  (a1,b3,a5,c4,e3,c2), (a1,b3,a5,c6,b4,c2), (a1,b3,a5,c6,d4,c2),
  (a1,b3,c5,e6,d4,c2), (a2,b4,c2,d4,e2,c1), (a2,b4,c6,d4,b3,c1),
  (a2,b4,c6,d4,e2,c1), (a2,b4,c6,e5,d3,c1), (a2,b4,d5,c3,e2,c1),
  (a2,b4,d5,f4,d3,c1), (a2,b4,d5,f4,e2,c1), (a2,c1,d3,f4,d5,c3),
  (a2,c1,e2,g3,e4,c3).

Crossrefs

Cf. A366778, A001230, A234623, A254129, A356404.

Sequence in context: A114317 A168320 A323134 * A081299 A117808 A002065

Adjacent sequences: A368496 A368497 A368498 * A368500 A368501 A368502

Keyword

nonn,hard,more

Author

Kaloyan Kapralov, Dec 27 2023

Status

approved