A001409
Number of 2n-step polygons on cubic lattice.
Original entry on oeis.org
1, 0, 3, 22, 207, 2412, 31754, 452640, 6840774, 108088232, 1768560270, 29764630632, 512705615350, 9005206632672, 160810554015408, 2912940755956084, 53424552150523386
Offset: 0
From _M. F. Hasler_, Jun 17 2025: (Start)
For n = 2, the three 4-step polygons are the 1 X 1 squares orthogonal to one of the three coordinate axes. (The sequence counts the polygons up to translations.)
For n = 3, the 22 six-step polygons can be partitioned into:
- six 2 X 1 rectangles (two in each of the previously considered planes);
- twelve L- or "seat" shaped polygons (as one can get by gluing together two 1 X 1 squares in a 90 degree angle along one side, or by folding a 2 X 1 rectangle by 90 degrees along the common side of its 1 X 1 square halves): choose one of the six half axes for the orientation of one of the squares, and one of the four orthogonal axes for the other, then divide by two because the order of the two choices doesn't matter;
- four polygons obtained by making three steps in direction of distinct axes (e.g., in direction of the three unit vectors) and then the same three steps in the opposite direction. The four inequivalent instances are obtained by rotating one of them three times by 90° around the same fixed axis. (End)
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor., 40 (2007), pp. 10973-11017, Table A5.
- G. S. Rushbrooke and J. Eve, High-temperature Ising partition function and related noncrossing polygons for the simple cubic lattice, J. Math. Physics, 3 (1962), pp. 185-189.
A337403
Table read by antidiagonals: T(w,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section w x w where the walk starts at the tube's edge.
Original entry on oeis.org
4, 12, 4, 36, 14, 4, 98, 54, 14, 4, 274, 200, 56, 14, 4, 702, 744, 224, 56, 14, 4, 1854, 2626, 926, 226, 56, 14, 4, 4614, 9186, 3738, 956, 226, 56, 14, 4, 11778, 31122, 15056, 4014, 958, 226, 56, 14, 4, 28914, 105766, 59092, 17074, 4050, 958, 226, 56, 14, 4
Offset: 1
T(1,2) = 12 as after a step in one of the two directions toward the adjacent tube edge 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 2*3 + 2*4 - 2 = 12.
The table begins:
4 12 36 98 274 702 1854 4614 11778 28914 72394 176310 435346...
4 14 54 200 744 2626 9186 31122 105766 351798 1175726 3859350 12729142...
4 14 56 224 926 3738 15056 59092 230254 881850 3367124 12712194 47952018...
4 14 56 226 956 4014 17074 71774 301578 1251362 5170636 21143094 86148002...
4 14 56 226 958 4050 17464 75060 325064 1400650 6040372 25882446 110668184...
4 14 56 226 958 4052 17506 75584 330070 1440668 6321926 27685144 121407404...
4 14 56 226 958 4052 17508 75632 330748 1447916 6386092 28180426 124857572...
4 14 56 226 958 4052 17508 75634 330802 1448768 6396174 28278426 125681952...
4 14 56 226 958 4052 17508 75634 330804 1448828 6397220 28292004 125825794...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397286 28293264 125843600...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293336 125845094...
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...
A338126
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 w apart where the walk starts on one of the planes.
Original entry on oeis.org
5, 20, 21, 80, 92, 93, 304, 392, 408, 409, 1168, 1684, 1832, 1852, 1853, 4348, 7036, 8084, 8308, 8332, 8333, 16336, 29396, 35752, 37620, 37936, 37964, 37965, 60208, 120776, 155756, 168768, 171808, 172232, 172264, 172265, 223352, 497196, 677856, 758340, 782344, 786972, 787520, 787556, 787557
Offset: 1
T(2,1) = 20 as after one step towards the opposite plane the walk must turn along that plane; this eliminates the 2-step straight walk in that direction, so the total number of walks is A116904(2) - 1 = 21 - 1 = 20.
The table begins:
5;
20,21;
80,92,93;
304,392,408,409;
1168,1684,1832,1852,1853;
4348,7036,8084,8308,8332,8333;
16336,29396,35752,37620,37936,37964,37965;
60208,120776,155756,168768,171808,172232,172264,172265;
223352,497196,677856,758340,782344,786972,787520,787556,787557;
817852,2026220,2920764,3379476,3545108,3586040,3592736,3593424,3593464,3593465;
A359133
Sum of square end-to-end displacements over all n-step self-avoiding walks of A359741.
Original entry on oeis.org
0, 6, 24, 78, 384, 8190, 8472, 178110, 193824, 4231662, 7072056, 102812142, 208526592, 2508914454, 5268441144, 62304671286, 124116667488, 1547651742990, 2850706506936, 38100453950670
Offset: 0
A140476
Number of self-avoiding walks on cubic lattice with no more than n steps.
Original entry on oeis.org
1, 7, 37, 187, 913, 4447, 21373, 102763, 490729, 2344615, 11154493, 53088643, 251931385, 1195905895, 5664817573, 26839963627, 126961839601, 600692091703, 2838415775797, 13414448995411, 63331776834145, 299041867336303
Offset: 0
a(9) = 1 + 6 + 30 + 150 + 726 + 3534 + 16926 + 81390 + 387966 + 1853886 = 2344615.
A330079
Number of n-step self-avoiding walks starting at the origin that are restricted to the boundary walls of the first octant of the cubic lattice.
Original entry on oeis.org
1, 3, 9, 27, 75, 213, 585, 1623, 4425, 12123, 32883, 89415, 241557, 653649, 1760427, 4747005, 12754593, 34301463, 91990575, 246880023, 661075149, 1771199169, 4736741853, 12673587057, 33856816431, 90482953989, 241499070195, 644781165933, 1719559634451, 4587222964881, 12225165127887
Offset: 0
A335307
The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where only the nodes have mass.
Original entry on oeis.org
1, 1, 1, 1, 5, 13, 31, 63, 141, 293, 665, 1553, 3795, 9225, 22257, 53623, 132277, 321651, 786553, 1928565, 4806503, 11885969, 29498995, 73362933, 184210629, 460165983, 1151961103
Offset: 1
a(1)-a(4) = 1 as the only stable walk is a walk straight down from the first node.
a(5) = 5. There are two stable walks with a first step to the right:
.
X-----+
| + X-----+
| | |
+-----+-----+ | |
| +-----+-----+
|
+
.
Assuming a node mass of p, both walks have a torque of 2p to the right and 2p to the left of the first node. These walks can be taken in 2 ways. Thus, with the straight down walk, the total number of stable walks is 2*2+1 = 5.
A335596
The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where only the connecting rods have mass.
Original entry on oeis.org
1, 1, 1, 1, 3, 7, 17, 43, 91, 183, 371, 799, 1941, 4621, 11463, 27823, 68997, 167481, 414045, 1006091, 2496981, 6127053, 15304071, 37838777, 95041475, 236320611, 595206771
Offset: 1
a(1)-a(4) = 1 as the only stable walk is a walk straight down from the first node.
a(5) = 3. There is one stable walk with a first step to the right:
.
X-----+
|
|
+-----+-----+-----+
,
Assuming a rod mass of q, the total torque to the right of the first node is 2*q*(1/2) + 1*q*1 = 2q. The total torque to the left of the first node is 1*q*(1/2) + 1*q*(3/2) = 2q. This walk can be taken in 2 ways. Thus, with the straight down walk, the total number of stable walks is 2+1 = 3.
A342807
Number of self-avoiding walks on a 3-dimensional cubic lattice where the walk consists of steps with incrementing length from 1 to n.
Original entry on oeis.org
1, 6, 30, 150, 750, 3750, 18630, 92406, 458262, 2270478, 11245590, 55697766, 275769654, 1365260862, 6758345838, 33450929886, 165549052326, 819248589606, 4054005363918
Offset: 0
a(1) to a(5) = 6*5^(n-1) as the number of walks equals the total number of non-backtracking walks when collisions are ignored.
a(6) = 18630 as, given one or more steps to the right followed by an upward step, the total number of walks that collide with a previous step is 5. These steps can be taking in 4*6 = 24 ways on the cubic lattice, giving 5*24 = 120 walks in all that are eliminated. The total number of walks ignoring collisions is 6*5^5 = 18750, so the total number of self-avoiding walks is 18750-120 = 18630.
A366058
Number of n-step self-avoiding walks on a 3D cubic lattice where no step is to a lattice point closer to the origin than the current point.
Original entry on oeis.org
1, 6, 30, 126, 462, 1566, 5070, 15966, 49422, 151326, 460110, 1392606, 4202382, 12656286, 38067150, 114398046, 343587342, 1031548446, 3096218190, 9291800286, 27881692302, 83657659806, 250998145230, 753044767326, 2259234965262, 6777906222366, 20334121320270, 61003169267166, 183011118414222
Offset: 0
a(2) = 30 as after two steps no walk can step closer to the origin than its current point, so a(2) = A001412(2) = 30.
a(3) = 126. Given the first two steps of the 3-step walk are to points (1,0,0) and (1,0,1) then a step to (0,0,1) is forbidden. This walk can be taken in 4*6 = 24 ways on the cubic lattice, so the total number of permitted walks is a(3) = A001412(3) - 24 = 150 - 24 = 126.
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