A116903 -id:A116903 - cp's OEIS frontend

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.

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

Scott R. Shannon, Sep 24 2021

This is a variation of A347990. The same walk rules apply except that the walk is confined to the upper two quadrants of the 2D square lattice. See A347990 for further details.

			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.

Cf. A347990 (four quadrants), A348009 (one quadrant), A116903, A001411, A337353.

A335098 The number of constructible vertically balanced self-avoiding walks of length n on the upper half-plane of a 2D square lattice where the nodes and connecting rods have equal mass.

Original entry on oeis.org

3, 5, 11, 23, 51, 109, 251, 549, 1291, 2981, 7067, 16571, 39601, 94195, 226997, 544687, 1320935, 3194399, 7797891, 18996977, 46651387, 114353905, 282109663, 694793903, 1720327219, 4253521985, 10565387267, 26213565665, 65300013637, 162516950805, 405892537979

Offset: 1

Scott R. Shannon, Sep 12 2020

This is a variation of A337860 where at every step, given the nodes and connecting rods have equal mass, the resulting 2D lattice structure is stable against toppling, assuming no sideways perturbations. See that sequence for further details of the allowed walks.

			a(1) = 3, a(2) = 5. These are the same stable walks as in A337860.
a(3) = 11. The constructible stable walks given a first step to the right are:
.
                                                   +
                        +      +---+   +---+       |
                        |      |           |       +
X---+---+---+   X---+---+  X---+       X---+       |
                                               X---+
.
These walks can also take a first step to the left thus, along with the directly vertical walk, the total number of stable walks is 2*5 + 1 = 11.
One 3-step walk which is not counted here, along with its parent 2-step walk, is:
.
+---+        +---+
|      ==>   |   |
X            X   +
.
After two steps the resulting structure is not stable against toppling, its center-of-mass is clearly to the right of the one node at y=0, thus any resulting 3-step walks resulting from this unstable 2-step walk are not counted.

Cf. A337860, A337317, A335780, A337761, A116903, A116904, A001411.

A336988 Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 2D square grid confined to an infinite strip of height 2h where the walk starts at coordinate (0,h).

Original entry on oeis.org

4, 10, 4, 22, 12, 4, 42, 34, 12, 4, 90, 82, 36, 12, 4, 182, 194, 98, 36, 12, 4, 382, 438, 262, 100, 36, 12, 4, 742, 1034, 650, 282, 100, 36, 12, 4, 1486, 2362, 1610, 754, 284, 100, 36, 12, 4, 2866, 5558, 3870, 1994, 778, 284, 100, 36, 12, 4

Offset: 1

Scott R. Shannon, Aug 10 2020

			T(1,3) = 22. The five 3-step walks taking a first step to the right and upward or a step upward and then to the right are:
.
      +  +--+     +--+  +--+--+  +--+
      |     |     |     |        |  |
X--+--+  X--+  X--+     X        X  +
.
The same steps can be taken to the right then down, to the left then down, and to the left then up. There is also the two straight walks right and left. This give a total number of walks of 4*5+2 = 22.
.
The table begins:
.
4 10 22  42  90 182  382  742  1486  2866   5646  10878  21198   40694   78758...
4 12 34  82 194 438 1034 2362  5558 12662  29366  66330 151566  339514  767798...
4 12 36  98 262 650 1610 3870  9490 22830  55826 134242 326934  784770 1901246...
4 12 36 100 282 754 1994 5046 12786 31746  79566 196858 491506 1214262 3024890...
4 12 36 100 284 778 2142 5682 14986 38462  98762 249894 635290 1599394 4048366...
4 12 36 100 284 780 2170 5882 15970 42286 111554 288962 748414 1916762 4921146...
4 12 36 100 284 780 2172 5914 16230 43730 117810 311894 823682 2146886 5593690...
4 12 36 100 284 780 2172 5916 16266 44058 119842 321630 862674 2284682 6040622...
4 12 36 100 284 780 2172 5916 16268 44098 120246 324394 877210 2348022 6281498...
4 12 36 100 284 780 2172 5916 16268 44100 120290 324882 880866 2368982 6380418...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324930 881446 2373706 6409762...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881498 2374386 6415746...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374442 6416534...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416594...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416596...

Cf. A336769 (start at origin), A001411 (h->infinity), A007825 (h=1), A116903, A038373, A336863, A336818.

For n <= h, T(h,n) = A001411(n).

Row 1 = T(1,n) = A007825(n).

A348010 Number of n-step self-avoiding walks on the upper half-plane of a 2D square lattice rotated by Pi/4.

Original entry on oeis.org

1, 2, 6, 14, 40, 96, 268, 664, 1820, 4588, 12464, 31712, 85704, 219376, 590640, 1518652, 4077112, 10518364, 28177388, 72883016, 194910964, 505202708, 1349189968, 3503014492, 9344407884, 24296044256, 64748290040, 168550939272

Offset: 0

Scott R. Shannon, Sep 24 2021

			The rotated lattice, where * is the origin and + are the lattice points, is:
      +       +       +       +
        \   /   \   /   \   /
          +       +       +
        /   \   /   \   /   \
      +       +       +       +
        \   /   \   /   \   /
     -----+-------*-------+------
.
a(1) = 2 as the only two steps available are the diagonal steps to the northeast and northwest of the origin.
a(2) = 6 as from each of the available first steps three steps are possible, giving a total of 2 * 3 = 6 steps.

A. J. Guttmann and G. M. Torrie, Critical behavior at an edge for the SAW and Ising model, J. Phys. A 17 (1984), 3539-3552.

Cf. A116903 (not rotated), A001411.

A348057 Number of n-step self-avoiding walks on three quadrants of a 2D square lattice.

Original entry on oeis.org

1, 4, 10, 28, 74, 202, 534, 1442, 3822, 10258, 27202, 72718, 192840, 514228, 1363342, 3629316, 9619264, 25575326, 67765590, 180001304, 476807826, 1265567600, 3351529410, 8890447682, 23538665948, 62409037914, 165202281046

Offset: 0

Scott R. Shannon, Sep 26 2021

			a(2) = 10. Assuming the lower left quadrant is the one removed then a walk of left-down or down-left is not permitted, so the total number of 2-step walks is 4 * 3 - 2 = 10.

Cf. A001411 (four quadrants), A116903 (two quadrants), A038373 (one quadrant), A129700 (half quadrant).

cp's OEIS Frontend

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.

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A335098 The number of constructible vertically balanced self-avoiding walks of length n on the upper half-plane of a 2D square lattice where the nodes and connecting rods have equal mass.

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A336988 Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 2D square grid confined to an infinite strip of height 2h where the walk starts at coordinate (0,h).

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A348010 Number of n-step self-avoiding walks on the upper half-plane of a 2D square lattice rotated by Pi/4.

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A348057 Number of n-step self-avoiding walks on three quadrants of a 2D square lattice.

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