A357500 Largest number of nodes of an induced path in the n X n knight graph.

1, 1, 7, 9, 15, 21, 24, 34

Offset

1,3

References

Thomas Dawson, Échecs Féeriques, L'Échiquier, volume 2, issue 2, 1930; issue 3, 1931.

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.

Links

Table of n, a(n) for n=1..8.

Nikolai Beluhov, Snake paths in king and knight graphs, arXiv:2301.01152 [math.CO], 2023.

Eric Weisstein's World of Mathematics, Knight Graph.

Wikipedia, Induced path

Formula

a(n) >= A165143(n) - 1 for n >= 3.

a(n) <= (4*(n^2+3*n-2)-1)/7 for n >= 4. - Elijah Beregovsky, Dec 22 2022

a(n) = n^2/2 + O(n) (Beluhov 2023). - Pontus von Brömssen, Jan 30 2023

Example

Longest paths for 3 <= n <= 7:
  X . X   . . . .   . X . X .   . X X X . .   X . X X X . .
  X . X   X . X X   X X . X X   X . . X X X   X . X . . X .
  X X X   . X X .   X . . . X   X . . . X X   . X X . . X .
          X X X X   X X . X X   X X . . . X   . . X . . X X
                    . X X X .   . X X . X X   . . . . . . X
                                . X X X X .   . X X X X X X
                                              X X . X X . .

Crossrefs

Cf. A165143, A331968.

Sequence in context: A112072 A024902 A111312 * A158891 A213220 A087680

Adjacent sequences: A357497 A357498 A357499 * A357501 A357502 A357503

Keyword

nonn,more

Author

Pontus von Brömssen, Oct 01 2022

Status

approved