cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A335806 The number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2x2x2 where the walk starts at the middle of the box's edge.

Original entry on oeis.org

1, 4, 12, 40, 118, 358, 936, 2600, 6212, 16068, 34936, 83708, 163452, 357056, 613592, 1205716, 1770616, 3073480, 3715920, 5573480, 5255048, 6591160, 4353912, 4330096, 1513712, 1061392, 0
Offset: 0

Views

Author

Scott R. Shannon, Aug 14 2020

Keywords

Examples

			a(1) = 4 as the walk is free to move one step in four directions.
a(2) = 12. A first step along either edge leading to the corner leaves two possible second steps. A first step to the centre of either face can be followed by a second step to three edges or to the center of the cube, four steps in all. Thus the total number of 2-step walks is 2*2+2*4 = 12.
a(26) = 0 as it is not possible to visit all 26 available lattice points when the walk starts from the middle of the box's edge.

Crossrefs

Cf. A336862 (other box sizes), A337021 (start at center of box), A337033 (start at center of face), A337034 (start at corner of box), A001412, A259808, A039648.

Formula

For n>=27 all terms are 0 as the walk contains more steps than there are available lattice points in the 2x2x2 box.

A337021 The number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2x2x2 where the walk starts at the center of the box.

Original entry on oeis.org

1, 6, 24, 72, 168, 456, 1032, 2712, 5784, 14640, 29760, 71136, 133344, 291696, 479232, 950880, 1343088, 2375808, 2774832, 4266240, 3909792, 5046672, 3230400, 3316704, 1122000, 808128, 0
Offset: 0

Views

Author

Scott R. Shannon, Aug 11 2020

Keywords

Examples

			a(1) = 6 as the walk is free to move one step in all six axial directions.
a(2) = 24 as after a step in one of the six axial directions the walk must turn along the face of the box; this eliminates the 2-step straight walk in all directions, so the total number of walks is 6*5-6 = 24.
a(26) = 0 as it is not possible to visit all 26 available lattice points when the walk starts from the middle of the box.

Crossrefs

Cf. A337023 (other box sizes), A337033 (start at center of face), A335806 (start at middle of edge), A337034 (start at corner of box), A001412, A039648.

Formula

For n>=27 all terms are 0 as the walk contains more steps than there are available lattice points in the 2x2x2 box.

A337031 Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2h x 2h x 2h where the walk starts at the center of one of the box's faces.

Original entry on oeis.org

5, 17, 5, 52, 21, 5, 148, 89, 21, 5, 400, 357, 93, 21, 5, 1060, 1424, 405, 93, 21, 5, 2700, 5484, 1789, 409, 93, 21, 5, 6720, 20960, 7705, 1849, 409, 93, 21, 5, 15760, 78412, 33048, 8257, 1853, 409, 93, 21, 5, 36248, 292168, 139032, 37097, 8329, 1853, 409, 93, 21, 5
Offset: 1

Views

Author

Scott R. Shannon, Aug 12 2020

Keywords

Examples

			T(1,2) = 17. Taking the first step right,left,forward or backward hits the box's edge after which the walks has three directions for the second step, giving 4*3 = 12 walks in all. A first step upward can be followed by a second step in five directions. The total number of 2-step walks is therefore 12+5 = 17.
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The table begins:
.
5 17 52 148  400 1060  2700   6720  15760   36248    77856   163296    312760...
5 21 89 357 1424 5484 20960  78412 292168 1072272  3919000 14145220  50832492...
5 21 93 405 1789 7705 33048 139032 583256 2422480 10053452 41415564 170419680...
5 21 93 409 1849 8257 37097 164533 728808 3194636 13978148 60739156 263711448...
5 21 93 409 1853 8329 37877 171117 776065 3496769 15758504 70593984 315942684...
5 21 93 409 1853 8333 37961 172165 786089 3577129 16326745 74257917 337994448...
5 21 93 409 1853 8333 37965 172261 787445 3591637 16455441 75254865 344977177...
5 21 93 409 1853 8333 37965 172265 787553 3593341 16475617 75451269 346633713...
5 21 93 409 1853 8333 37965 172265 787557 3593461 16477709 75478437 346921841...
5 21 93 409 1853 8333 37965 172265 787557 3593465 16477841 75480957 346957465...
5 21 93 409 1853 8333 37965 172265 787557 3593465 16477845 75481101 346960453...
5 21 93 409 1853 8333 37965 172265 787557 3593465 16477845 75481105 346960609...
5 21 93 409 1853 8333 37965 172265 787557 3593465 16477845 75481105 346960613...

Crossrefs

Cf. A116904 (h->infinity), A337033 (h=1), A337023 (start at center of box), A336862 (start at middle of edge), A337035 (start at corner of box), A001412.

Formula

For n <= h, T(h,n) = A116904(n).
Row 1 = T(1,n) = A337033(n).
For n >= (2h+1)^3, T(h,n) = 0 as the walk contains more steps than there are available lattice points in the 2h X 2h X 2h box.

A337034 The number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2x2x2 where the walk starts at one of the box's corners.

Original entry on oeis.org

1, 3, 9, 30, 96, 294, 840, 2214, 5796, 14112, 34158, 76062, 167928, 337476, 670626, 1181064, 2067900, 3103404, 4666542, 5758008, 7176144, 6899904, 6743712, 4535916, 3117192, 1098900, 392628, 0
Offset: 0

Views

Author

Scott R. Shannon, Aug 12 2020

Keywords

Examples

			a(1) = 3 as the walk can take a first step in only three directions along the cube's edges.
a(3) = 9. After the first step along the cube's edge the walk can turn toward a face center in two ways. From the face center is has four available directions. If instead the walk takes two steps along the cube's edge to another corner it then has only two directions available for a third step. As the first step can be taken in three ways the total number of 3-step walks is 3*2*4+3*2 = 30.
a(26) = 392628. This is the total number of ways a 26-step walk can completely fill the 2x2x2 box's 26 available lattice points. Unlike the walk which starts at the center of the box, see A337021, all lattice points can be visited in one walk. Note this is larger than the total number of walks when starting at the center of the cube's face, see A337033.

Crossrefs

Cf. A337035 (other box sizes), A337021 (start at center of box), A337033 (start at center of face), A335806 (start at middle of edge), A001412, A039648.

Formula

For n>=27 all terms are 0 as the walk contains more steps than there are available lattice points in the 2x2x2 box.
Showing 1-4 of 4 results.

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