A337035 -id:A337035 - cp's OEIS frontend

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.

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

Scott R. Shannon, Aug 11 2020

			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..

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.

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.

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

Scott R. Shannon, Aug 14 2020

			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...

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.

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.

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

Scott R. Shannon, Aug 12 2020

			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:
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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...

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.

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

Scott R. Shannon, Aug 12 2020

			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.

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.

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

cp's OEIS Frontend

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.

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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.

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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.

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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.

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