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.

A337023 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 the box.

Original entry on oeis.org

6, 24, 6, 72, 30, 6, 168, 144, 30, 6, 456, 624, 150, 30, 6, 1032, 2520, 720, 150, 30, 6, 2712, 9360, 3408, 726, 150, 30, 6, 5784, 34008, 15432, 3528, 726, 150, 30, 6, 14640, 120960, 68088, 16776, 3534, 726, 150, 30, 6, 29760, 430656, 289128, 79320, 16920, 3534, 726, 150, 30, 6
Offset: 1

Views

Author

Scott R. Shannon, Aug 11 2020

Keywords

Examples

			T(1,2) = 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.
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The table begins:
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6 24  72 168  456  1032  2712   5784   14640   29760    71136    133344    291696..
6 30 144 624 2520  9360 34008 120960  430656 1511856  5340312  18587208  65176416..
6 30 150 720 3408 15432 68088 289128 1205976 4920528 19985928  80066136 321160728..
6 30 150 726 3528 16776 79320 366960 1677864 7516992 33312456 145379760 630249720..
6 30 150 726 3534 16920 81216 385224 1822584 8518920 39588480 181800312 829567656..
6 30 150 726 3534 16926 81384 387768 1850376 8765304 41478144 194837136 912538512..
6 30 150 726 3534 16926 81390 387960 1853664 8805504 41872944 198158520 937459176..
6 30 150 726 3534 16926 81390 387966 1853880 8809632 41928816 198761160 942984312..
6 30 150 726 3534 16926 81390 387966 1853886 8809872 41933880 198836352 943868424..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934144 198842448 943966968..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842736 943974192..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974504..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974510..

Crossrefs

Cf. A001412 (h->infinity), A337021 (h=1), A337031 (start at center of face), A337035 (start as corner of box), A336862 (start at middle of edge), A116904, A039648.

Formula

For n <= h, T(h,n) = A001412(n).
Row 1 = T(1,n) = A337021(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.

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.

A337033 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 one of the box's faces.

Original entry on oeis.org

1, 5, 17, 52, 148, 400, 1060, 2700, 6720, 15760, 36248, 77856, 163296, 312760, 590536, 995160, 1663664, 2405056, 3482320, 4180656, 5080320, 4823560, 4686432, 3165088, 2228584, 792272, 303264, 0
Offset: 0

Views

Author

Scott R. Shannon, Aug 12 2020

Keywords

Examples

			a(1) = 5 as the walk is free to move one step in five possible directions. It cannot take a step to a direction opposite to the face's normal it starts on.
a(2) = 17. Taking the first along the starting face 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 away from the starting face can be followed by a second step in five directions. The total number of 2-step walks is therefore 12+5 = 17.
a(26) = 303264. 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.

Crossrefs

Cf. A337031 (other box sizes), A337021 (start at center of box), A335806 (start at middle of edge), A337034 (start at corner of box), A001412, A116904.

Formula

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