A358083
Sum of square end-to-end displacements over all n-step self-avoiding walks of A358046.
Original entry on oeis.org
4, 16, 128, 448, 2256, 5376, 29424, 69888, 302568, 741376, 3026448, 7216896, 29268352, 65785216, 263892736, 591065568, 2279452040, 5195776064, 19324558176, 44442289024, 161417689504, 371206519136, 1328055630144, 3044451252064, 10774811055304, 24625495784320, 86363375773808, 197092099990080
Offset: 1
a(3) = 128 as, in the first quadrant, the four 3-step SAWs that have the first and last visited lattice point being mutually visible are:
.
X
|
X---. . .---X X
| | | |
X---. X---. X---. X---.---.
.
The sum of square end-to-end displacements of these four walks is 1 + 5 + 5 + 5 = 16. They can be walked in eight different ways on a square lattice thus a(3) = 16 * 8 = 128.
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.
A359073
Sum of square end-to-end displacements over all n-step self-avoiding walks of A359709.
Original entry on oeis.org
0, 4, 16, 44, 160, 556, 1744, 12252, 15840, 98876, 138160, 709900, 1155616, 5098260, 11820656, 37085908, 111147104, 281078764, 932893104, 2255139900, 7295211968, 18928121236, 54864568720, 160016686500, 404167501888, 1331607134172, 2945597090384, 10805511468852, 21448743511648
Offset: 0
A359709
Number of n-step self-avoiding walks on a 2D square lattice whose end-to-end distance is an integer.
Original entry on oeis.org
1, 4, 4, 12, 28, 76, 164, 732, 1044, 4924, 6724, 30636, 43972, 190516, 313996, 1197908, 2284260, 7678188, 16257604, 50524252, 113052396, 341811828, 773714436, 2358452388, 5245994292, 16447462492, 35395532236, 115129727188, 238542983748, 804980005276
Offset: 0
a(3) = 12 as, in the first quadrant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):
.
X---.
|
X---.
.
This can be walked in 8 different ways on a 2D square lattice. There are also the four walks directly along the x and y axes, giving a total of 8 + 4 = 12 walks.
A368614
Number of n-step self-avoiding walks on a 2D square lattice where each visited lattice point is either a neighbor of the first visited lattice point, else the first visited lattice point is directly visible (cf. A358036) from the lattice point when it is first visited.
Original entry on oeis.org
4, 8, 16, 24, 48, 80, 168, 296, 624, 1144, 2424, 4552, 9680, 18480, 39368, 76128, 162376, 317288, 677624, 1335688, 2856536, 5672576, 12149080, 24280768, 52079424, 104665200, 224825088, 454047672, 976721744, 1981083216, 4267578200, 8689274768, 18743542208, 38295782400, 82715689712
Offset: 1
a(4) = 24. For walks with a second step in the first quadrant, there are three 4-step saws where the first lattice point is either a neighbor or directly visible from each point as it is first visited. These are:
.
.---.---. .---. .
| | |
X---. . .
| |
X---. .
|
X---.
.
where 'X' marks the position of the first lattice point. These three walks can be taken in eight ways on the 2D square lattice, so the total number of walks is 3 * 8 = 24.
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