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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 by 1 after each step up until the step length is n. No closed-loop path is possible until n = 7.

Like A010566 all possible paths are counted, including those that are equivalent via rotation and reflection.

For n = 8, 15, 20, 24, 27, 32, 35, 39, 44, ... = A380867, the path can be a rectangle. The first two cases are illustrated through the "Images" link from Scott R. Shannon. These numbers correspond to triangular numbers T(n) for which there are n1 > n2 > n3 > n4 >= 0 such that T(n) = 2(A+B) for A = T(n1) - T(n2) = T(n3) - T(n4) and B = T(n2) - T(n3). See A380867 for more. - M. F. Hasler, Mar 14 2025

Consider a self-avoiding walk on a 2D square lattice where two visited lattice points are considered to be visible from each other if, on drawing a line directly between these two points, the line neither crosses another lattice point which has been visited by previous steps of the walk, nor crosses any line directly connecting two consecutively visited lattice points that forms a part of the path of the walk. This sequence lists the number of n-step self-avoiding walks for which the first visited lattice point of the walk is directly visible from the last visited point. See the examples below.

For the 29-step walk the ratio of the number of end-to-end visible walks to all walks is a(29)/A001411(29) = 1730939615552/6279396229332 ~ 0.276. The value and behavior of this ratio as n -> infinity is unknown.

See A358046 for the number of walks when only the visited lattice points are considered when determining point visibility.

Consider a self-avoiding walk on a 2D square lattice where two visited lattice points are considered to be visible from each other if either no other lattice points exist on the line drawn directly between these two lattice points, or if such points exist, they have not been visited by previous steps of the walk. This sequence lists the number of n-step self-avoiding walks for which the first visited lattice point of the walk is directly visible from the last visited point. See the examples below.

For the walks studied there is a difference in the ratio for the number of end-to-end visible walks to all walks for steps with even-n to odd-n. For example a(28)/A001411(28) ~ 0.72, while a(29)/A001411(29) ~ 0.88. The values and behavior of these ratios as n -> infinity is unknown.

See A358036 for the number of walks where the path between lattice points is also considered when determining point visibility.

This sequence gives the number of self-avoiding walks on a 2-dimensional square lattice where the walk consists of steps of length 1 to n which can be taken in any order. Walks which visit the same lattice coordinates but are done so by taking steps of the same length in different order are considered to be different walks. For example a walk consisting of steps with length 1 and 2 to the right is counted as a different walk to one with step lengths 2 and 1 to the right.

The first time a collision with a previous step can occur is for n = 4. If we only consider the first step being taken to the right then there are six ways this can occur. These are 2R->3U->1L->4D, 3R->1U->2L->4D, 3R->2U->1L->4D, 4R->1U->2L->3D, 4R->1U->3L->2D, 4R->2U->1L->3D, where the number is the step length and R,L,U,D are directions right,left,up and down from the origin.

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.

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.

This sequence gives the number of self-avoiding walks on a 2-dimensional 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.

The first time a collision with a previous step can occur is for n = 8, i.e., a walk with step lengths of 1,4,9,16,25,36,49,64. For a walk with one or more initial steps to the right followed by an upward step this can occur in nine different ways. For example, consider a walk with steps of length 1,4,9,16,25 to the right, a step of length 36 upward, then a step of length 49 to the left. A step of length 64 downward would now result in a collision. Requiring eight steps before a collision is in contrast to the standard 2D square lattice SAW of A001411 where a collision can occur on the fourth step.

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.

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.