A334602 -id:A334602 - cp's OEIS frontend

A335305 Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps of length 1 to n which can be taken in any order.

0, 0, 0, 0, 0, 0, 16128, 287232, 0, 0, 1843367680, 45589291776, 0, 0

Offset: 1

Scott R. Shannon, May 31 2020

This sequence gives the number of closed-loop self avoiding walks on a 2D square lattice where the walk consists of steps of length 1 to n which can be taken in any order. No closed-loop path is possible until n = 7.
As in A334720 the only n values which can form closed loops are those which correspond to even triangular numbers; any path must take the same number of steps back toward the origin as it does away from the origin in each of the four possible directions to form a closed loop, so the total sum of the steps in these directions must be even. As the walks consist of the steps of length 1 to n this implies only walks for which the sum of 1 to n is even can form closed loops.
Like A010566 all possible paths are counted, including those that are equivalent via rotation and reflection. Also counted as different walks are loops which visit identical lattice points but are done so by taking steps in a different order. This leads to an extremely rapid increase in the total number of loops possible as n increases.
a(15) is currently unknown but is likely to be about 6*10^15.

			a(1) to a(6) = 0 as no closed loop paths are possible.
a(7) = 16128. Given the first step has length 1 and is to the right, with the next non-right step being upward, there are 84 different loops. Each of these can be walked in at least 2 ways, with the single perfect square having 48 different possible walks. Each of these in turn can be started with a first step of length 1 to n, and each of these can then be walked in 8 different ways on a 2D square grid. This gives a total number of 7-step paths of 16128. This should be compared with A334720 where for n=7 only 8 paths are possible. See the attached link text file for more details of n=7.

A. R. Conway et al., Algebraic techniques for enumerating self-avoiding walks on the square lattice, J. Phys A 26 (1993) 1519-1534.
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, Details of the number of loops for n=7.

Cf. A334720, A334877, A334602, A000217, A010566, A002931, A334756.

A336262 Number of self-avoiding walks on a 2-dimensional square lattice where the walk consists of steps with incrementing lengths equal to the prime numbers, from 2 to prime(n).

Original entry on oeis.org

1, 4, 12, 36, 108, 324, 972, 2876, 8364, 24124, 69116, 196916, 559604, 1585764, 4495740, 12714796, 35654620, 99686708, 278880060, 781504972, 2180418716, 6079373324, 16857930068, 46773551052, 129562831140, 358157148332

Offset: 0

Scott R. Shannon, Jul 15 2020

The first time a collision with a previous step can occur is for n = 7, i.e., a walk with steps of length 2,3,5,7,11,13,17. If we consider only walks starting with one or more steps to the right followed by an upward step then a collision can occur in five ways. These are 2R->3U->5U->7U->11R->13D->17L, 2R->3U->5U->7U->11L->13D->17R, 2R->3U->5R->7R->11U->13L->17D, 2R->3U->5R->7R->11D->13L->17U, 2R->3R->5R->7R->11U->13L->17D, where the number is the step length and R,L,U,D are directions right,left,up and down on the grid. Requiring seven steps before a collision can occur is in contrast to the walk where the steps' lengths increment by 1, see A334877, which requires only six steps.

			a(1) = 4. These are the four ways one can step away from the origin on a 2D square lattice.
a(2) = 12. These consist of the two following walks:
.
        *
        |
        .
        | 3        2         3
        .      *---.---*---.---.---*
        |
*---.---*
     2
.
The first walk can be taken in eight different ways on the 2D square lattice, the second in four ways, giving a total of 12 walks.
a(7) = 2876. If we consider only walks starting with one or more steps to the right followed by an upward step, and ignoring collisions, then the total number of walks is 3^5+3^4+3^3+3^2+3^1+3^0 = 364. However, five of these are forbidden due to the collisions given in the comments, leaving 359 in total. These can be walked in eight different ways on the 2D grid. There are also the four straight walks along the axes. This gives a total of 359*8+4 = 2876 walks.

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.

Cf. A334877, A001411, A334602, A334720, A077482, A334720.

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).

Original entry on oeis.org

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.

A342807 Number of self-avoiding walks on a 3-dimensional cubic lattice where the walk consists of steps with incrementing length from 1 to n.

Original entry on oeis.org

1, 6, 30, 150, 750, 3750, 18630, 92406, 458262, 2270478, 11245590, 55697766, 275769654, 1365260862, 6758345838, 33450929886, 165549052326, 819248589606, 4054005363918

Offset: 0

Scott R. Shannon, Mar 22 2021

This sequence gives the number of self-avoiding walks on a 3-dimensional 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. The first time a collision with a previous step can occur is for n = 6. See A334877 for further details.

			a(1) to a(5) = 6*5^(n-1) as the number of walks equals the total number of non-backtracking walks when collisions are ignored.
a(6) = 18630 as, given one or more steps to the right followed by an upward step, the total number of walks that collide with a previous step is 5. These steps can be taking in 4*6 = 24 ways on the cubic lattice, giving 5*24 = 120 walks in all that are eliminated. The total number of walks ignoring collisions is 6*5^5 = 18750, so the total number of self-avoiding walks is 18750-120 = 18630.

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

Cf. A334877, A342800, A001412, A334602, A336262.

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A335305 Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps of length 1 to n which can be taken in any order.

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A336262 Number of self-avoiding walks on a 2-dimensional square lattice where the walk consists of steps with incrementing lengths equal to the prime numbers, from 2 to prime(n).

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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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A342807 Number of self-avoiding walks on a 3-dimensional cubic lattice where the walk consists of steps with incrementing length from 1 to n.

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