A352388 -id:A352388 - cp's OEIS frontend

A353176 Maximum length of a finite snake in the Snake Number Problem with n-periodic instructions in an infinite square grid (see Comments).

30, 79, 152, 450, 241, 257, 1098, 1448, 9520, 8804, 8338, 11348, 25316, 18823

Offset: 5

Rodolfo Kurchan, Apr 28 2022

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. The snake is either finite (if it gets stuck at some point) or infinite (if it can go on forever). a(n) is the maximum number of squares visited by a finite snake. For n <= 4, all snakes are infinite. - Pontus von Brömssen, May 08 2022

			a(5) = 30
URDDL: 30
-- -- 20 21 -- --
-- 18 19 22 23 --
16 17 02 03 24 --
15 14 01 04 25 26
12 13 06 05 30 27
11 10 07 -- 29 28
-- 09 08 -- -- --
.
a(6) = 79
UURDLL: 79
-- -- -- -- -- -- -- 74 75 -- --
-- -- -- -- -- 69 70 73 76 77 --
-- -- -- 64 65 68 71 72 79 78 --
-- -- -- 63 66 67 -- 27 28 -- --
-- -- 61 62 -- 22 23 26 29 30 --
56 57 60 17 18 21 24 25 32 31 --
55 58 59 16 19 20 03 04 33 34 --
54 53 52 15 14 13 02 05 06 35 36
-- -- 51 -- -- 12 01 08 07 38 37
-- -- 50 49 48 11 10 09 40 39 --
-- -- -- -- 47 -- 43 42 41 -- --
-- -- -- -- 46 45 44 -- -- -- --
.
   n | Number of finite solutions | Maximum length | Instructions that give
     |         A352388(n)         |      a(n)      |   the maximum length
  -------------------------------------------------------------------------
   5                    5                  30         URDDL
   6                   21                  79         UURDLL
   7                  127                 152         URULLDD
   8                  618                 450         URURUULD
   9                 2934                 241         URRRRDLRR
  10                13542                 257         URRLDLRRUR
  11                61803                1098         URUURUUULLD
  12               276650                1448         URUULLDUDDDD
  13              1219508                9520         URRRLLDLRRULL
  14              5309179                8804         URRURRRLDLRULL
  15             22868295                8338         UDDRULUUUULLULD
  16             97663066               11348         URRURRRLDDLRUULL
  17            414156142               25316         URRRDLULUUUUULURL
  18           1746438478               18823         UDDDRULULLULLUULDU
Computer solutions a(5) to a(13) found by Giorgio Vecchi.
Computer solutions a(14) to a(18) found by Ariel Futoransky.

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.

Cf. A352388, A353259, A353060, A028444.

A353234 Maximum length of 2 finite snakes in the Snake Number Problem with n-periodic instructions in an infinite square grid (see Comments).

Original entry on oeis.org

32, 44, 138, 226, 326, 310, 409, 1138, 5265, 10499

Offset: 4

Rodolfo Kurchan, May 01 2022

We start with 2 infinite snakes, and 4 possible directions: up, right, down, left.
If on its turn one of the snakes cannot execute an order because that square is occupied, it goes to the next order, and so on.
The snakes can be blocked and finish there or can continue infinitely.
Which are the longest finite snakes using n instructions for 2 snakes that start in the same square (with n >= 4 because with 3 or fewer instructions are infinite)?

			4 INSTRUCTIONS
URDL: 32
   --  30  31  --  --  --
   25  26  32   7   8  --
   24  20   2   3   9  13
   17  18   1   4  15  14
   16  12   6   5  19  21
   --  11  10  29  23  22
   --  --  --  28  27  --
   1) Both snakes start in square 1.
   2) First snake go U = 2, second snake can't go Up.
   3) First snake goes R = 3.
   4) Second snake goes R = 4.
   5) First snake cannot go D, so second snake D = 5.
   6) First snake cannot go L, so second snake goes L = 6
   7) First snake can go U = 7, and second cannot go U.
   8) First snake can go R = 8, second snake cannot go R.
   9) First snake can go D = 9, second snake can go D = 10.
  10) First snake cannot go L, second snake can go L = 11.
  11) First snake cannot go U, second snake can go U = 12.
  12) First snake can go R = 13, second snake cannot go R.
  13) First snake can go D = 14, second snake cannot go D.
  14) First snake can go L = 15, second snake can go L = 16.
And so on.
                      | Instructions that give
   n | Maximum length |   the maximal length
  --------------------------------------------
   4          32              URDL
   5          44              URDLU
   6         138              UURDLL
   7         226              UURDUUL
   8         326              UURDLUUL
   9         310              UUUURDUUL
  10         409              URUDLLDURR
  11        1138              UDDRDDRDUDL
  12        5265              URUDRDURDDDL
  13       10499              UUDRUDDLUULDD
Computer solutions found by Giorgio Vecchi.

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.

Cf. A353259, A353060, A352388.

cp's OEIS Frontend

A353176 Maximum length of a finite snake in the Snake Number Problem with n-periodic instructions in an infinite square grid (see Comments).

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