A352388 - OEIS

%I #56 May 07 2022 09:56:03

%S 0,0,0,0,5,21,127,618,2934,13542,61803,276650,1219508,5309179,

%T 22868295,97663066,414156142,1746438478

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

%C 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

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

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

%H Ariel Futoransky, <a href="http://snake.puzzlefun.online/">Snake Program</a>, Snake Program to try the snakes, April 2022 (see bottom animation and label for different options).

%H Rodolfo Kurchan, <a href="http://www.puzzlefun.online/problems">Puzzle Fun</a>, Snake Number Problem, March 2022.

%e These are the 5 different solutions with 5 instructions:

%e UURDL: 19

%e   17 18  3  4 --

%e   16 19  2  5  6

%e   15 14  1  8  7

%e   -- 13 10  9 --

%e   -- 12 11 -- --

%e URDLL: 21

%e   -- 19 20 -- -- --

%e   17 18 21  2  3 --

%e   16 15 14  1  4  5

%e   -- 12 13  8  7  6

%e   -- 11 10  9 -- --

%e URRDL: 24

%e   -- -- 20 21 22

%e   17 18 19 24 23

%e   16 15  2  3  4

%e   13 14  1  6  5

%e   12 11  8  7 --

%e   -- 10  9 -- --

%e URDLU: 26

%e   -- -- 21 22 --

%e   16 17 20 23 24

%e   15 18 19 26 25

%e   14 13  2  3 --

%e   -- 12  1  4  5

%e   -- 11 10  7  6

%e   -- --  9  8 --

%e URDDL: 30

%e   -- -- 20 21 -- --

%e   -- 18 19 22 23 --

%e   16 17  2  3 24 --

%e   15 14  1  4 25 26

%e   12 13  6  5 30 27

%e   11 10  7 -- 29 28

%e   --  9  8 -- -- --

%Y Cf. A353259, A353060, A353176, A004147.

%K nonn,more

%O 1,5

%A _Rodolfo Kurchan_, Apr 29 2022