A001412 -id:A001412 - cp's OEIS frontend

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.

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.
.
The table begins:
.
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.

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

Scott R. Shannon, Aug 11 2020

			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.

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.

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

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.

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

Scott R. Shannon, Aug 12 2020

			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.

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.

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

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.

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

Scott R. Shannon, Aug 12 2020

			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.
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The table begins:
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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...

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.

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.

A010567 Number of 2n-step self-avoiding closed paths (or cycles) on the 3-dimensional cubic lattice.

Original entry on oeis.org

6, 24, 264, 3312, 48240, 762096, 12673920, 218904768, 3891176352, 70742410800, 1309643747808, 24609869536800, 468270744898944, 9005391024862848, 174776445357365040, 3419171337633496704

Offset: 1

N. J. A. Sloane

This sequence agrees with A001413 except for n=1, for which the given value is "purely conventional" (although the convention is non-standard): it counts 6 two-step closed paths, all of which visit no node twice but use an edge twice, so whether they are "self-avoiding" is indeed a matter of agreement. Same considerations apply to the first terms of A010568-A010570. - Andrey Zabolotskiy, May 29 2018

M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.

Essentially the same as A001413.

Cf. A010568 (analog in 4 dimensions), A010569 (in 5D), A010570 (in 6D), A130706 (in 1D), A010566 (in 2D, different convention for n=1), A002896 (closed walks, not necessarily self-avoiding), A001412 (self-avoiding walks, not necessarily closed), A039618, A038515.

def A010567(n): # For illustration - becomes slow for n > 5
    if not hasattr(A:=A010567, 'terms'):
        A.terms=[6]; O=0,; A.paths=[(O*3, (1,)+O*2, t+O)for t in((2,0),(1,1))]
    while n > len(A.terms):
        for L in (0,1):
            new = []; cycles = 0
            for path in A.paths:
                end = path[-1]
                for i in (0,1,2):
                   for s in (1,-1):
                      t = tuple(end[j]if j!=i else end[j]+s for j in (0,1,2))
                      if t not in path: new.append(path+(t,))
                      elif L and t==path[0]: cycles += 24 if path[2][1] else 6
            A.paths = new
        A.terms.append(cycles)
    return A.terms[n-1] # M. F. Hasler, Jun 17 2025

a(8)-a(10) copied from A001413 by Andrey Zabolotskiy, May 29 2018
a(11)-a(12) copied from A001413 by Pontus von Brömssen, Feb 28 2024
a(13)-a(16) (using A001413) from Alois P. Heinz, Feb 28 2024
Name edited and "self-avoiding" added by M. F. Hasler, Jun 17 2025

A337401 Triangle read by rows: T(n,w) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section 2w X 2w where the walk starts at the center of the tube's side.

Original entry on oeis.org

5, 19, 21, 72, 91, 93, 258, 383, 407, 409, 926, 1638, 1821, 1851, 1853, 3176, 6856, 8019, 8295, 8331, 8333, 11000, 28810, 35506, 37531, 37921, 37963, 37965, 36988, 119106, 155492, 168399, 171691, 172215, 172263, 172265, 125302, 492766, 683126, 758182, 781811, 786823, 787501, 787555, 787557

Offset: 1

Scott R. Shannon, Aug 26 2020

			T(2,1) = 19 as after a step in one of the two directions toward the adjacent tube side the walk must turn along the side; this eliminates the 2-step straight walk in those two directions, so the total number of walks is 4*4 + 5 - 2 = 19.
The table begins:
5;
19,21;
72,91,93;
258,383,407,409;
926,1638,1821,1851,1853;
3176,6856,8019,8295,8331,8333;
11000,28810,35506,37531,37921,37963,37965;
36988,119106,155492,168399,171691,172215,172263,172265;
125302,492766,683126,758182,781811, 786823,787501,787555,787557;
414518,2013142,2981996,3393526,3545117,3585297,3592551,3593403,3593463,3593465;

Scott R. Shannon, Data for n=1 to n=14.

Cf. A337400 (start at middle of tube), A337403 (start at tube's edge), A116904 (w->infinity), A001412, A337023, A259808, A039648.

For w>=n, T(n,w) = A116904(n).

A338125 Triangle read by rows: T(n,w) is the number of n-step self avoiding walks on a 3D cubic lattice confined between two infinite planes a distance 2w apart where the walk starts at the middle point between the planes.

Original entry on oeis.org

6, 28, 30, 124, 148, 150, 516, 692, 724, 726, 2156, 3196, 3492, 3532, 3534, 8804, 14324, 16428, 16876, 16924, 16926, 36388, 64076, 76956, 80700, 81332, 81388, 81390, 148452, 282716, 354740, 380964, 387052, 387900, 387964, 387966, 609812, 1251044, 1631420, 1795212, 1843452, 1852716, 1853812, 1853884, 1853886

Offset: 1

Scott R. Shannon, Oct 11 2020

			T(2,1) = 28 as after a step in one of the two directions towards the planes the walk must turn along the plane; this eliminates the 2-step straight walk in those two directions, so the total number of walks is A001412(2) - 2 = 30 - 2 = 28.
The table begins:
6;
28,30;
124,148,150;
516,692,724,726;
2156,3196,3492,3532,3534;
8804,14324,16428,16876,16924,16926;
36388,64076,76956,80700,81332,81388,81390;
148452,282716,354740,380964,387052,387900,387964,387966;
609812,1251044,1631420,1795212,1843452,1852716,1853812,1853884,1853886;
2478484,5493804,7431100,8377908,8712892,8795020,8808420,8809796,8809876,8809878;

Scott R. Shannon, Full data table for n=1 to n=15.

Cf. A338126 (start on a plane), A001412 (w->infinity), A001412, A337023, A337400, A039648.

For w>=n, T(n,w) = A001412(n).

A359741 Number of n-step self-avoiding walks on a 3D cubic lattice whose end-to-end distance is an integer.

Original entry on oeis.org

1, 6, 6, 30, 78, 1134, 1350, 20574, 23238, 390606, 496998, 7614750, 10987926, 152120934, 237122526, 3110708214, 5017927638, 64718847438, 105210653478, 1362453235998

Offset: 0

Scott R. Shannon, Jan 12 2023

The walks counted are all those directly along and x, y or z axes, and all walks whose final (x,y,z) lattice point is a solution to the Pythagorean quadruple x^2 + y^2 + z^2 = t^2. The first such solution with all coordinates > 0 is 1^2 + 2^2 + 2^2 = 3^2, which explains the large increase in the number of walks from a(4) to a(5).

			a(3) = 30 as, in the first octant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):
.
     X---.
         |
     X---.
.
This can be walked in 24 different ways on a 3D cubic lattice. There are also the six walks directly along the x, y and z axes, giving a total of 24 + 6 = 30 walks.

Wikipedia, Self-avoiding walk.
Wikipedia, Pythagorean quadruple.

Cf. A359133, A001412, A359709, A118313.

cp's OEIS Frontend

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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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.

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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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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.

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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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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.

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A010567 Number of 2n-step self-avoiding closed paths (or cycles) on the 3-dimensional cubic lattice.

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A337401 Triangle read by rows: T(n,w) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section 2w X 2w where the walk starts at the center of the tube's side.

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A338125 Triangle read by rows: T(n,w) is the number of n-step self avoiding walks on a 3D cubic lattice confined between two infinite planes a distance 2w apart where the walk starts at the middle point between the planes.

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A359741 Number of n-step self-avoiding walks on a 3D cubic lattice whose end-to-end distance is an integer.

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