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.

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.

A337035 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 h x h x h where the walk starts at one of the box's corners.

Original entry on oeis.org

3, 6, 3, 12, 9, 3, 18, 30, 9, 3, 30, 96, 33, 9, 3, 24, 294, 120, 33, 9, 3, 18, 840, 456, 123, 33, 9, 3, 0, 2214, 1662, 486, 123, 33, 9, 3, 0, 5796, 6018, 1908, 489, 123, 33, 9, 3, 0, 14112, 20784, 7584, 1944, 489, 123, 33, 9, 3, 0, 34158, 70470, 29754, 7932, 1947, 489, 123, 33, 9, 3
Offset: 1

Views

Author

Scott R. Shannon, Aug 12 2020

Keywords

Examples

			T(2,3) = 30. 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.
.
The table begins:
.
3 6 12  18  30   24   18     0      0      0       0        0        0         0...
3 9 30  96 294  840 2214  5796  14112  34158   76062   167928   337476    670626...
3 9 33 120 456 1662 6018 20784  70470 231648  754386  2396832  7562730  23297826...
3 9 33 123 486 1908 7584 29754 115866 444096 1678560  6260082 23037330  84061494...
3 9 33 123 489 1944 7932 32298 132720 541908 2212542  8946288 36007908 143452686...
3 9 33 123 489 1947 7974 32766 136590 570570 2397384 10062258 42243138 176723826...
3 9 33 123 489 1947 7977 32814 137196 576168 2443284 10386522 44376156 189622260...
3 9 33 123 489 1947 7977 32817 137250 576930 2451066 10456566 44914830 193454916...
3 9 33 123 489 1947 7977 32817 137253 576990 2452002 10467042 45017580 194310204...
3 9 33 123 489 1947 7977 32817 137253 576993 2452068 10468170 45031314 194456058...
3 9 33 123 489 1947 7977 32817 137253 576993 2452071 10468242 45032652 194473668...
3 9 33 123 489 1947 7977 32817 137253 576993 2452071 10468245 45032730 194475234...
3 9 33 123 489 1947 7977 32817 137253 576993 2452071 10468245 45032733 194475318...
3 9 33 123 489 1947 7977 32817 137253 576993 2452071 10468245 45032733 194475321...

Crossrefs

Cf. A039648 (h->infinity), A337034 (h=2), A337031 (start at center of face), A337032 (start as center of box), A336862 (start at middle of edge), A001412.

Formula

For n <= h, T(h,n) = A039648(n).
Row 2 = T(2,n) = A337034(n).
For n >= (h+1)^3, T(h,n) = 0 as the walk contains more steps than there are available lattice points in the hxhxh box.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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