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.

A336862 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 middle of the box's edge.

Original entry on oeis.org

4, 12, 4, 40, 14, 4, 118, 54, 14, 4, 358, 208, 56, 14, 4, 936, 826, 224, 56, 14, 4, 2600, 3232, 936, 226, 56, 14, 4, 6212, 12688, 3862, 956, 226, 56, 14, 4, 16068, 48924, 16196, 4026, 958, 226, 56, 14, 4, 34936, 187276, 67346, 17246, 4050, 958, 226, 56, 14, 4
Offset: 1

Views

Author

Scott R. Shannon, Aug 14 2020

Keywords

Examples

			T(1,2) = 12. A first step along either edge leading to the corner leaves two possible second steps. A first step to the center of either face can be followed by a second step to three edges or to the center of the box, four steps in all. Thus the total number of 2-step walks is 2*2+2*4 = 12.
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The table begins:
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4 12 40 118 358  936  2600  6212  16068   34936   83708   163452    357056...
4 14 54 208 826 3232 12688 48924 187276  705196 2627950  9670620  35231628...
4 14 56 224 936 3862 16196 67346 282676 1180326 4950936 20646098  86165926...
4 14 56 226 956 4026 17246 73588 316456 1358518 5860464 25266192 109288486...
4 14 56 226 958 4050 17478 75288 327778 1425340 6236152 27260378 119641050...
4 14 56 226 958 4052 17506 75600 330362 1444544 6360718 28020896 123963354...
4 14 56 226 958 4052 17508 75632 330766 1448280 6391426 28238732 125405300...
4 14 56 226 958 4052 17508 75634 330802 1448788 6396618 28285548 125766436...
4 14 56 226 958 4052 17508 75634 330804 1448828 6397242 28292536 125835068...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397286 28293288 125844228...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293336 125845120...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293338 125845172...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293338 125845174...

Crossrefs

Cf. A259808 (h->infinity), A335806 (h=1), A337023 (start at center of box), A337031 (start at center of face), A337035 (start at corner of box), A001412, A039648.

Formula

For n <= h, T(h,n) = A259808(n).
Row 1 = T(1,n) = A335806(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.

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.

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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