A335305 -id:A335305 - cp's OEIS frontend

A336265 Number of 2D closed-loop self-avoiding paths on a square lattice where each path consists of steps with successive lengths equal to the prime numbers, from 2 to prime(2n+1).

0, 0, 0, 0, 0, 0, 56, 64, 448, 1552, 8952, 65120, 284584, 1491800, 8467816, 48961856, 307751136, 1781258728

Offset: 0

Scott R. Shannon, Jul 15 2020

This sequence gives the number of closed-loop self avoiding walks on a 2D square lattice where the walk consists of steps with successive lengths equal to the prime numbers. No closed loop path is possible until n = 6, i.e. prime(13) = 41. This walk consists of steps of length 2,3,5,7,11,13,17,19,23,29,31,37,41.

Similar to A010566, where only an even number of steps can form a closed loop, here only an odd number can. This is due to the requirement that the total distance stepped in each of the four directions away from the origin must be matched by an equal distance in the opposite direction. As all primes, other than 2, are odd and unique, this can only occur if the total number of steps in a given direction (other than the direction of the first step of length 2) is even. However the first single step of length 2 still requires an even number of odd length steps to return to the origin, giving an odd number of steps overall in that direction. Adding up the four directions gives an overall odd number for the total number of steps.

			a(0) to a(5) = 0 as no closed-loop walk is possible.
a(6) = 56. There are seven walks which form closed loops when considering only those which start with one or more steps to the right followed by a step upward. These walks consist of steps with lengths 2,3,5,7,11,13,17,19,23,29,31,37,41. See the attached linked text file for the images. Each of these can be walked in eight ways on a 2D square lattice, giving a total number of closed loops of 7*8 = 56.
See the attached linked text files for images of n = 7 and n = 8.

A. J. Guttmann, On the critical behavior of self-avoiding walks, J. Phys. A 20 (1987), 1839-1854.
A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.
Scott R. Shannon, Images for closed-loops for n = 6, maximum prime = 41.
Scott R. Shannon, Images for closed-loops for n = 7, maximum prime = 47.
Scott R. Shannon, Images for closed-loops for n = 8, maximum prime = 59.

Cf. A010566, A334720, A335305, A334877, A334602.

A342800 Number of self-avoiding polygons on a 3-dimensional cubic lattice where each walk consists of steps with incrementing length from 1 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 24, 72, 0, 0, 1704, 5184, 0, 0, 193344, 600504, 0, 0, 34321512, 141520752, 0, 0, 9205815672, 37962945288, 0, 0

Offset: 1

Scott R. Shannon, Mar 21 2021

This sequence gives the number of self-avoiding polygons (closed-loop self-avoiding walks) on a 3D cubic lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. Like A334720 and A335305 only n values corresponding to even triangular numbers can form closed loops. All possible paths are counted, including those that are equivalent via rotation and reflection.

			a(1) to a(6) = 0 as no self-avoiding closed-loop walk is possible.
a(7) = 24 as there is one walk which forms a closed loop which can be walked in 24 different ways on a 3D cubic lattice. These walks, and those for n(8) = 72, are purely 2-dimensional. See A334720 for images of these walks.
a(11) = 1704. These walks consist of 120 purely 2-dimensional walks and 1584 3-dimensional walks. One of these 3-dimensional walks is:
.
                                /|
                               / |                        z  y
                              /  |                        | /
                        7 +y /   |                        |/
                            /    | 8 -z                   |----- x
             6 +x          /     |
  |---.---.---.---.---.---/      |               9 +x
  |                              |---.---.---.---.---.---.---.---.---/
  | 5 +z                                                            /
  |                                                                /
  |---.---.---.---/                                               /
        4 -x     /  3 +y                                         /
                /                                               /  10 -y
                | 2 +z                                         /
                |                                             /
                | 1 +z                                       /
                X---.---.---.---.---.---.---.---.---.---.---/
                                     11 -x
.

A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

Cf. A334720, A334877, A342807, A001413, A002896, A002899, A010566, A335305.

A345676 Number of closed-loop self-avoiding paths on a 2-dimensional square lattice where each path consists of steps with successive lengths equal to the square numbers, from 1 to n^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 368, 264, 0, 0, 1656, 5104, 0, 0, 62016, 105344, 0, 0, 1046656, 3181104

Offset: 1

Scott R. Shannon, Sep 04 2021

This sequence gives the number of closed-loop self-avoiding walks on a 2D square lattice where the walk starts with a step length of 1 which then increments at each step to the next square number until the step length is n^2. No closed-loop path is possible until n = 15.
Like A334720 and A335305 the only n values that can form closed loop walks are those which correspond to the indices of even triangular numbers. Curiously though n = 16 walks form no closed loops, even though both n = 15 and n = 16 are indices of such numbers.
As in A010566 all possible paths are counted, including those that are equivalent via rotation and reflection.

			a(1) to a(14) = 0 as no closed-loop paths are possible.
a(15) = 32 as there are four different paths which form closed loops, and each of these can be walked in eight different ways on a 2D square lattice. These walks consist of steps with lengths 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. See the linked text images.

A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.
Scott R. Shannon, Images of the closed loops for n = 15. The line lengths in this text file are long so it may need to be downloaded to be viewed correctly.

Cf. A347506, A334720, A336265, A337550, A010566, A000217.

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A336265 Number of 2D closed-loop self-avoiding paths on a square lattice where each path consists of steps with successive lengths equal to the prime numbers, from 2 to prime(2n+1).

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A342800 Number of self-avoiding polygons on a 3-dimensional cubic lattice where each walk consists of steps with incrementing length from 1 to n.

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A345676 Number of closed-loop self-avoiding paths on a 2-dimensional square lattice where each path consists of steps with successive lengths equal to the square numbers, from 1 to n^2.

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