A352388 Number of solutions to Snake Number Problem for snakes with n-periodic instructions in an infinite square grid (see Comments).

0, 0, 0, 0, 5, 21, 127, 618, 2934, 13542, 61803, 276650, 1219508, 5309179, 22868295, 97663066, 414156142, 1746438478

Offset: 1

Author

Rodolfo Kurchan, Apr 29 2022

Keywords

• nonn
• more

Comments

Given a list of n move instructions (up, right, down, left), the snake starts at the origin and moves according to the instructions, in order. If an instruction tells it to move to a square that has already been visited, the snake skips that instruction. After it has followed (or skipped) the last instruction in the list, it starts again with the first one. a(n) is the number of lists of n instructions that results in the snake getting stuck at some point. Lists of instructions that are equivalent under rotations and reflections are counted only once, so we can for example assume that the first instruction is "up", and that the first "right" comes before the first "left". But how does one know when to interrupt a snake and deem it infinite? - Pontus von Brömssen, May 05 2022

Computer solutions a(5) to a(13) found by Giorgio Vecchi.

Computer solution a(14) to a(18) found by Ariel Futoransky.

Examples

			These are the 5 different solutions with 5 instructions:
UURDL: 19
  17 18  3  4 --
  16 19  2  5  6
  15 14  1  8  7
  -- 13 10  9 --
  -- 12 11 -- --
URDLL: 21
  -- 19 20 -- -- --
  17 18 21  2  3 --
  16 15 14  1  4  5
  -- 12 13  8  7  6
  -- 11 10  9 -- --
URRDL: 24
  -- -- 20 21 22
  17 18 19 24 23
  16 15  2  3  4
  13 14  1  6  5
  12 11  8  7 --
  -- 10  9 -- --
URDLU: 26
  -- -- 21 22 --
  16 17 20 23 24
  15 18 19 26 25
  14 13  2  3 --
  -- 12  1  4  5
  -- 11 10  7  6
  -- --  9  8 --
URDDL: 30
  -- -- 20 21 -- --
  -- 18 19 22 23 --
  16 17  2  3 24 --
  15 14  1  4 25 26
  12 13  6  5 30 27
  11 10  7 -- 29 28
  --  9  8 -- -- --
		

Links

• Ariel Futoransky, Snake Program, Snake Program to try the snakes, April 2022 (see bottom animation and label for different options).
• Rodolfo Kurchan, Puzzle Fun, Snake Number Problem, March 2022.

Crossrefs

Cf. A353259, A353060, A353176, A004147.