A360917 Array read by antidiagonals: T(m,n) is the number of vertices in the longest induced path in the grid graph P_m X P_n.

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 6, 8, 9, 9, 8, 6, 7, 9, 11, 11, 11, 9, 7, 8, 11, 14, 14, 14, 14, 11, 8, 9, 12, 16, 17, 17, 17, 16, 12, 9, 10, 14, 18, 20, 21, 21, 20, 18, 14, 10, 11, 15, 20, 22, 24, 24, 24, 22, 20, 15, 11, 12, 17, 22, 25, 27, 29, 29, 27, 25, 22, 17, 12

Offset: 1

Author

Andrew Howroyd, Feb 26 2023

Keywords

• nonn
• tabl

Comments

Equivalently, T(m,n) is the maximum number of unit squares of a snake-like polyomino in an m X n rectangle.

Examples

			Array begins:
==============================================
  m\n|  1  2  3  4  5  6  7  8  9 10 11 12 ...
-----+----------------------------------------
   1 |  1  2  3  4  5  6  7  8  9 10 11 12 ...
   2 |  2  3  5  6  8  9 11 12 14 15 17 18 ...
   3 |  3  5  7  9 11 14 16 18 20 22 24 26 ...
   4 |  4  6  9 11 14 17 20 22 25 28 30 33 ...
   5 |  5  8 11 14 17 21 24 27 30 34 37 40 ...
   6 |  6  9 14 17 21 24 29 32 36 40 44 47 ...
   7 |  7 11 16 20 24 29 33 38 42 46 50 55 ...
   8 |  8 12 18 22 27 32 38 42 48 52 57 62 ...
   9 |  9 14 20 25 30 36 42 48 53 58 64 70 ...
  10 | 10 15 22 28 34 40 46 52 58 64 71 77 ...
  11 | 11 17 24 30 37 44 50 57 64 71 77 86 ...
  12 | 12 18 26 33 40 47 55 62 70 77 86 92 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..435
• Nikolai Beluhov, Snake paths in king and knight graphs, arXiv:2301.01152 [math.CO], 2023.
• Eric Weisstein's World of Mathematics, Grid Graph.

Crossrefs

Main diagonal is A331968.

Cf. A360199, A360915, A360916 (maximum induced paths), A360920.

Formula

T(m,n) = T(n,m).

T(m,n) = 2*m*n/3 + O(m+n) (Beluhov 2023, Proposition 3). - Pontus von Brömssen, May 08 2023