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.
A347990
Number of n-step self-avoiding walks on a 2D square lattice where the walk cannot step to the smaller square ring of numbers than the ring it is currently on.
Original entry on oeis.org
1, 4, 12, 36, 92, 252, 628, 1644, 4052, 10340, 25332, 63708, 155452, 387036, 941948, 2328740, 5657236, 13914596, 33757804, 82713164, 200467108, 489746916, 1186060492, 2891000036, 6997192716, 17025058164, 41186981772, 100070851212, 242000513660, 587312389940
Offset: 0
a(0..3) are the same as the standard square lattice SAW of A001411 as the walk cannot step to a smaller ring in the first three steps.
a(4) = 92. If we restrict the first one or more steps to the right followed by an upward step then there is one walk which steps to a smaller ring and is thus forbidden. That is the walk (0,0) -> (1,0) -> (2,0) -> (2,1) -> (1,1). As this can be walked in eight different ways on the square lattice the number of 4-step walks becomes A001411(4) - 8 = 100 - 8 = 92.
A337550
Number of closed-loop self-avoiding paths of length 4n on a 2D square lattice where no step can be in the same direction as the previous step.
Original entry on oeis.org
8, 0, 24, 64, 360, 1728, 8624, 43776, 225216, 1173280, 6182704, 32905536, 176657000, 955629920, 5204178360, 28509374976, 157005901896, 868756900608, 4827586102216, 26929911745600, 150750954809952, 846588050093632, 4768197762850608
Offset: 1
a(1) = 8. The single walk of length 4 is:
.
+---+
| |
+---+
.
This can be taken in 8 different ways on a square lattice, giving a total 1*8 = 8.
a(2) = 0 as there is no closed-loop path consisting of 8 steps.
a(3) = 24. There is one walk, ignoring reflection and rotations, with a length of 12. The walk is:
.
+---+
| |
+---+ +---+
| |
+---+ +---+
| |
+---+
.
This can be walked in 3 different ways if the first steps are right and then upward. This path can be then taken in 8 ways on a square lattice, giving a total number of 3*8 = 24.
a(4) = 64. There is one walk, with indistinct reflections and rotations, with a length of 16. The walk is:
.
+---+
| |
+---+ +---+
| |
+---+ +---+
| |
+---+ +---+
| |
+---+
.
This can be walked in 8 different ways if the first steps are right and then upward. This path can be then taken in 8 ways on a square lattice, giving a total number of 8*8 = 64.
.
a(5) = 360. There are four walks, with indistinct reflections and rotations, with a length of 20. The walks, and the different ways they can be taken, are:
.
+---+ +---+
| | | |
+---+ +---+ +---+ +---+
| | | |
+---+ +---+ +---+ +---+
| | | |
+---+ +---+ +---+ +---+
| | | |
+---+ +---+ +---+ +---+
| | x 10 | | x 20
+---+ +---+
+---+ +---+
| | | |
+---+ +---+ +---+ +---+
| | | |
+---+ +---+ +---+ +---+
| | | |
+---+ +---+ +---+ +---+
| | | |
+---+ +---+ +---+ +---+
| | x 5 | | x 10
+---+ +---+
.
Each of these can be walked in 8 different ways on a square lattice, giving a total number of 8*(10+20+5+10) = 360.
See the attached text file for images of the closed-loops for n=1 to n=11.
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.
A348008
Number of n-step self-avoiding walks on the upper two quadrants of a 2D square lattice where the walk cannot step to the smaller square ring of numbers than the ring it is currently on.
Original entry on oeis.org
1, 3, 7, 19, 45, 115, 273, 683, 1629, 4035, 9643, 23713, 56761, 138883, 332807, 811343, 1945777, 4730655, 11351999, 27542291, 66123953, 160174529, 384700337, 930720767, 2236106651, 5404679299, 12988762401, 31370201873, 75409375419, 182019777165, 437648513199
Offset: 0
a(0..3) are the same as the standard SAW on the upper two quadrants of a square lattice, see A116903, as the walk cannot step to a smaller ring in the first three steps.
a(4) = 45. If we restrict the first one or more steps to the right followed by an upward step then there is one walk which steps to a smaller ring and is thus forbidden. That is the walk (0,0) -> (1,0) -> (2,0) -> (2,1) -> (1,1). As this can be walked in four different ways in the upper two quadrants the number of 4-step walks becomes A116903(4) - 4 = 49 - 4 = 45.
A348009
Number of n-step self-avoiding walks on one quadrant of a 2D square lattice where the walk cannot step to the smaller square ring of numbers than the ring it is currently on.
Original entry on oeis.org
1, 2, 4, 10, 22, 52, 118, 282, 646, 1544, 3576, 8546, 19924, 47612, 111536, 266488, 626520, 1496670, 3528470, 8427952, 19913078, 47559756, 112572916, 268857568, 637327742, 1522153378, 3612811784, 8629110414, 20503211908, 48975965026, 116478744692
Offset: 0
a(0..3) are the same as the standard SAW on one quadrant of a square lattice, see A038373, as the walk cannot step to a smaller ring in the first three steps.
a(4) = 22. If we restrict the first one or more steps to the right followed by an upward step then there is one walk which steps to a smaller ring and is thus forbidden. That is the walk (0,0) -> (1,0) -> (2,0) -> (2,1) -> (1,1). As this can be walked in two different ways in one quadrant the number of 4-step walks becomes A038373(4) - 2 = 24 - 2 = 22.
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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