A358036
Number of n-step self-avoiding walks on a 2D square lattice where the first visited lattice point is directly visible from the last visited lattice point, and were both the visited lattice points and the path between these points are considered when determining the visibility of points.
Original entry on oeis.org
0, 8, 24, 48, 144, 336, 992, 2344, 6760, 16336, 46432, 113904, 320864, 793136, 2222824, 5524040, 15409704, 38493560, 106895408, 268253720, 742053704, 1869175480, 5154271008, 13022699248, 35816428904, 90722285632, 248960813992, 631978627880, 1730939615552
Offset: 1
a(1) = 0 as after one step in any of the four available directions the first and last point of the walk are directly connected by a line forming the path, so are not considered mutually visible.
a(2) = 8 as there are 4*3 = 12 2-step SAWs, but the four walks which consist of two steps directly along the axes have a visited lattice point directly between the first and last points of the walk, so those two point are not visible from each other. Thus a(2) = 12 - 4 = 8.
a(3) = 24 as there are thirty-six 3-step SAWs which include four walks directly along the axes which have a first point that is not visible from the last. In the first quadrant there is one other walk whose second-step path is intersected by the line between the first and last points of the walk. This walk is:
.
.---X
|
X---.
.
where the first and last points are shown as 'X'. The above walk can be walked in eight ways on the 2D square lattice, so the total number of walks where the first point is visible from the last is 36 - 4 - 1*8 = 36 - 12 = 24.
a(4) = 48 as there are one hundred 4-step SAWs which include four walks directly along the axes which have a first point that is not visible from the last. In the first quadrant there are six other walks which have either previously visited points directly on the line between the first and last points of the walk, or in which this line intersects the path of previous steps. These walks are:
.
X .---X X
| | |
@---. @ @---. .---.---X .---. .---X
| | | | | | |
X---. X---. X---. X---. X---@ X X---.---.
.
where the visited points on the line between the first and last points are shown as '@'. Each of the above walks can be walked in eight ways on the 2D square lattice, so the total number of walks where the first point is visible from the last is 100 - 4 - 6*8 = 100 - 52 = 48.
A358046
Number of n-step self-avoiding walks on a 2D square lattice where the first visited lattice point is directly visible from the last visited lattice point, and were only visited lattice points are considered when determining the visibility of points.
Original entry on oeis.org
4, 8, 32, 64, 240, 480, 1904, 3832, 13992, 29304, 103088, 219416, 765600, 1609176, 5611680, 11785240, 40641032, 86254960, 293015872, 628547128, 2108574592, 4556118936, 15143701888, 32875906992, 108521571624, 236390241280, 776007097296, 1695412485136, 5538287862344
Offset: 1
a(1) = 4 as after one step in any of the four available directions the lattice point stepped to and the starting point have no other points between them, so the first point is visible from the last for all four walks.
a(2) = 8 as there are 4*3 = 12 2-step SAWs, but the four walks which consist of two steps directly along the axes have a visited lattice point directly between the first and last points of the walk, so those two point are not visible from each other. Thus a(2) = 12 - 4 = 8.
a(3) = 32 as there are thirty-six 3-step SAWs, and of those, only the four walks directly along the axes have visited points between the first and last points, so a(3) = 36 - 4 = 32.
a(4) = 64 as there are one hundred 4-step SAWs which include four walks directly along the axes which have a first point that is not visible from the last. In the first quadrant there are four other walks which have points on the line between the first and last point, and these points have been visited by earlier steps. These walks are:
.
X .---X X
| | |
@---. @ @---. .---.
| | | | |
X---. X---. X---. X---@ X
.
where the first and last points are shown as 'X' and where the visited points on the line between these two points are shown as '@'. Each of the above walks can be walked in eight ways on the 2D square lattice, so the total number of walks where the first point is visible from the last is 100 - 4 - 4*8 = 100 - 36 = 64.
A347506
Number of self-avoiding walks on a 2-dimensional square lattice where the walk consists of steps with incrementing lengths equal to the square numbers, from 1 to n^2.
Original entry on oeis.org
1, 4, 12, 36, 108, 324, 972, 2916, 8676, 25572, 74124, 213788, 614444, 1757012, 5001372, 14175996, 40113156, 113363284, 319328028, 897533236, 2521069708, 7052715556, 19742289948, 55129924484, 153874225436
Offset: 0
a(1) = 4. These are the four directions one can step away from a point on a 2D square lattice.
a(2) = 12. These consist of the two following walks:
.
*
|
.
|
. 4
| 1 4
. *---*---.---.---.---*
|
*---*
1
.
The first walk can be taken in 8 different ways, the second in 4 ways, giving a total of 12 walks.
a(8) = 8676. 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 3^6 + 3^5 + 3^4 + 3^3 + 3^2 + 3^1 + 3^0 = 1093. However, nine of these are forbidden due to the collisions given in the comments, leaving 1084 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 1084*8 + 4 = 8676 walks.
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
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.
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