A001413 -id:A001413 - cp's OEIS frontend

A001412 Number of n-step self-avoiding walks on cubic lattice.

1, 6, 30, 150, 726, 3534, 16926, 81390, 387966, 1853886, 8809878, 41934150, 198842742, 943974510, 4468911678, 21175146054, 100121875974, 473730252102, 2237723684094, 10576033219614, 49917327838734, 235710090502158, 1111781983442406, 5245988215191414, 24730180885580790, 116618841700433358, 549493796867100942, 2589874864863200574, 12198184788179866902, 57466913094951837030, 270569905525454674614

Offset: 0

N. J. A. Sloane

Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 5.10, pp. 331-339.
B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 462.
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).

R. D. Schram, G. T. Barkema, and R. H. Bisseling, Table of n, a(n) for n = 0..36
N. Clisby, Enumerative combinatorics of lattice polymers, Notices AMS, 68:4 (2021), 504-515. (Excellent survey)
N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. 40 (2007), 10973-11017, Table A5 for n<=30.
Steven R. Finch, Self-Avoiding-Walk Connective Constants
M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
A. J. Guttmann, On the critical behavior of self-avoiding walks, J. Phys. A 20 (1987), 1839-1854.
B. J. Hiley and M. F. Sykes, Probability of initial ring closure in the restricted random-walk model of a macromolecule, J. Chem. Phys., 34 (1961), 1531-1537.
D. S. McKenzie and C. Domb, The second osmotic virial coefficient of athermal polymer solutions, Proceedings of the Physical Society, 92 (1967) 632-649.
A. M. Nemirovsky, Karl F. Freed, Takao Ishinabe, and Jack F. Douglas, Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
D. Randall, Counting in Lattices: Combinatorial Problems from Statistical Mechanics, PhD Thesis (1994).
Raoul D. Schram, Gerard T. Barkema, and Rob H. Bisseling, Exact enumeration of self-avoiding walks, arXiv:1104.2184 [math-ph], 2011.
Nobu C. Shirai and Naoyuki Sakumichi, Negative Energetic Elasticity of Lattice Polymer Chain in Solvent, arXiv:2202.12483 [cond-mat.soft], 2022.
M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62.
M. F. Sykes, Self-avoiding walks on the simple cubic lattice, J. Chem. Phys., 39 (1963), 410-411.
M. F. Sykes, A. J. Guttmann, M. G. Watts, and P. D. Roberts, The asymptotic behavior of self-avoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.
M. F. Sykes, D. S. McKenzie, M. G. Watts, and J. L. Martin, The number of self-avoiding rings on a lattice, J. Phys. A 5 (1972), 661-666.

Cf. A002902, A078717, A001411, A001413.

mo = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}}; a[0] = 1;
a[tg_, p_: {{0, 0, 0}}] := Block[{e, mv = Complement[Last[p] + # & /@ mo, p]},
If[tg == 1, Return[Length@mv],Sum[a[tg - 1, Append[p, e]], {e, mv}]]];
a /@ Range[0, 8]
(* Robert FERREOL, Nov 30 2018, after the program of Giovanni Resta in A001411 *)

def add(L,x):
    M=[y for y in L];M.append(x)
    return(M)
plus=lambda L,M : [x+y for x,y in zip(L,M)]
mo=[[1,0,0],[-1,0,0],[0,1,0],[0,-1,0],[0,0,1],[0,0,-1]]
def a(n,P=[[0,0,0]]):
    if n==0: return(1)
    mv1 = [plus(P[-1],x) for x in mo]
    mv2=[x for x in mv1 if x not in P]
    if n==1: return(len(mv2))
    else: return(sum(a(n-1,add(P,x)) for x in mv2))
[a(n) for n in range(8)]
# Robert FERREOL, Nov 30 2018

A002902 Number of n-step self-avoiding walks on a cubic lattice with a first step along the positive x, y, or z axis.

Original entry on oeis.org

3, 15, 75, 363, 1767, 8463, 40695, 193983, 926943, 4404939, 20967075, 99421371, 471987255, 2234455839, 10587573027, 50060937987, 236865126051, 1118861842047, 5288016609807, 24958663919367, 117855045251079, 555890991721203, 2622994107595707

Offset: 1

N. J. A. Sloane

B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 462.
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).

D. S. McKenzie and C. Domb, The second osmotic virial coefficient of athermal polymer solutions, Proceedings of the Physical Society, 92 (1967) 632-649.
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
M. F. Sykes, Self-avoiding walks on the simple cubic lattice, J. Chem. Phys., 39 (1963), 410-411.
M. F. Sykes et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.

Equals (1/2)*A001412. Cf. A078717, A001411, A001413.

Name amended by Scott R. Shannon, Sep 17 2020

A038515 Number of self-avoiding walks on a tetrahedral (diamond) net, having 2n steps and forming a closed loop.

Original entry on oeis.org

1, 0, 0, 6, 12, 120, 564, 4074, 25008, 174618, 1181100, 8358306, 59167872, 427081512, 3103408308, 22797207330, 168616517760, 1256350493196

Offset: 0

Wouter Meeussen

Anthony J. Guttmann and Iwan Jensen, Appendix: Series Data and Growth Constant, Amplitude and Exponent Estimates, in: Polygons, Polyominoes and Polycubes, LNP 775, Springer, 2009. See Table 16.12 on p. 480 for the sequence a(n)/n.
Sean A. Irvine, Java program (github)

Cf. A001394, A001667, A001413, A001337, A344070, A344071.

a(0)=1 and a(9)-a(15) from Sean A. Irvine, Jan 16 2021

a(16)-a(17) using the data from Guttmann & Jensen added by Andrey Zabolotskiy, Jun 02 2022

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

A010568 Number of 2n-step self-avoiding closed paths on the 4-dimensional cubic lattice.

Original entry on oeis.org

8, 48, 912, 22944, 652320, 20266368, 669756192, 23146172544, 827460518688, 30378237727200, 1139447186954208, 43501453488658368, 1685588678025512832

Offset: 1

N. J. A. Sloane

From M. F. Hasler, Jun 18 2025: (Start)
This sequence gives the number of self-avoiding paths starting and ending at the origin. It does not consider equivalence with respect to translations, rotations or reflections. So it does count multiple cycles whose set of edges represents the same polygon. For example, a(2) = 48 counts 4-step cycles which all correspond to a unit square.
This explains why 8n | a(n) for all n, and a(n)/8n = (1, 3, 38, 717, 16308, 422216, 11959932, 361658946, 11492507204, 379727971590, ...)
Also, a(n = 1) = 8 gives the 2-step cycles which correspond to one step in any of the 8 directions, and one step back. Although this path is self-avoiding since it does not cross any point multiple times, it uses the same edge for the first and second step, and therefore might be excluded from the counting. In three dimensions this is done in A001413, A001413(1) = 0, as opposed to A010567(1) = 6. For two dimensions, this alternate definition is also used in A010566(1) = 0.
(End)

Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. 40 (2007), 10973-11017. Table A6 "Enumeration results for d = 4", column p_n, row 2*n gives a(n)/(4*n) for n>1.
Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion. [Tables in machine-readable format on separate pages.]
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.

Cf. A010566, A010567, A010569, A010570 (similar for dimension 2 through 6).

def A010568(n): # For illustration - becomes slow for n > 5
    if not hasattr(A:=A010568, 'r'):
       A.terms = [8]; O = 0,; I = O*4, (1,*O*3)
       A.paths = (*I, (2,*O*3)), (*I, (1,1,0,0))
    while n > len(A.terms):
        for L in (0, 1):
            new=[]; cycles = 0; O=(0,)*4; I = 0,1,2,3
            for path in A.paths:
                end = path[-1]; weight = 48 if path[2][1] else 8
                for i in I:
                   for s in (1, -1):
                      t = tuple(end[j]if j!=i else end[j]+s for j in I)
                      if t not in path: new.append(path+(t,))
                      elif L and t==O: cycles += weight
            A.paths = new
        A.terms.append(cycles)
    return A.terms[n-1] # M. F. Hasler, Jun 17 2025

For all n, a(n) is divisible by 8*n. - M. F. Hasler, Jun 18 2025

a(6)-a(8) from Sean A. Irvine, May 31 2018
a(9)-a(10) from Sean A. Irvine, Aug 09 2020
"Self-avoiding" inserted in definition by M. F. Hasler, Jun 18 2025
a(11)-a(13) from Clisby et al.'s data added by Andrei Zabolotskii, Jun 25 2025

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

N. J. A. Sloane

			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.

Cf. A001412, A001413.

More terms from R. J. Mathar, Aug 31 2007

A001667 2n-step polygons on b.c.c. lattice.

Original entry on oeis.org

96, 1776, 43776, 1237920, 37903776, 1223681760, 41040797376, 1416762272736, 50027402384640, 1799035070369856

Offset: 2

N. J. A. Sloane

Number of 2n-step closed self-avoiding walks starting from the origin. - Bert Dobbelaere, Jan 16 2019

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).

P. Butera and M. Comi, Enumeration of the self-avoiding polygons on a lattice by the Schwinger-Dyson equations, Annals of Combinatorics 3, 277-286 (1999); arXiv:cond-mat/9903297, 1999.
M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
M. F. Sykes et al., The number of self-avoiding walks on a lattice, J. Phys. A 5 (1972), 661-666.
Index entries for sequences related to b.c.c. lattice

Cf. A001666, A001413, A001337, A038515.

a(9)-a(10) from Bert Dobbelaere, Jan 16 2019

a(11) from Butera & Comi added by Andrey Zabolotskiy, Jun 02 2022

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

Jonathan Vos Post, Jun 29 2008

nonn

Primes include a(1) = 7, a(2) = 37, a(5) = 4447, a(8) = 102763, a(15) = 26839963627.

			a(9) = 1 + 6 + 30 + 150 + 726 + 3534 + 16926 + 81390 + 387966 + 1853886 = 2344615.

Partial sums of A001412.

Cf. A001411, A001412, A001413, A002902, A078717.

A342800 Number of self-avoiding polygons on a 3-dimensional cubic lattice where each walk consists of steps with incrementing length from 1 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 24, 72, 0, 0, 1704, 5184, 0, 0, 193344, 600504, 0, 0, 34321512, 141520752, 0, 0, 9205815672, 37962945288, 0, 0

Offset: 1

Scott R. Shannon, Mar 21 2021

This sequence gives the number of self-avoiding polygons (closed-loop self-avoiding walks) on a 3D cubic lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. Like A334720 and A335305 only n values corresponding to even triangular numbers can form closed loops. All possible paths are counted, including those that are equivalent via rotation and reflection.

			a(1) to a(6) = 0 as no self-avoiding closed-loop walk is possible.
a(7) = 24 as there is one walk which forms a closed loop which can be walked in 24 different ways on a 3D cubic lattice. These walks, and those for n(8) = 72, are purely 2-dimensional. See A334720 for images of these walks.
a(11) = 1704. These walks consist of 120 purely 2-dimensional walks and 1584 3-dimensional walks. One of these 3-dimensional walks is:
.
                                /|
                               / |                        z  y
                              /  |                        | /
                        7 +y /   |                        |/
                            /    | 8 -z                   |----- x
             6 +x          /     |
  |---.---.---.---.---.---/      |               9 +x
  |                              |---.---.---.---.---.---.---.---.---/
  | 5 +z                                                            /
  |                                                                /
  |---.---.---.---/                                               /
        4 -x     /  3 +y                                         /
                /                                               /  10 -y
                | 2 +z                                         /
                |                                             /
                | 1 +z                                       /
                X---.---.---.---.---.---.---.---.---.---.---/
                                     11 -x
.

A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

Cf. A334720, A334877, A342807, A001413, A002896, A002899, A010566, A335305.

cp's OEIS Frontend

A001412 Number of n-step self-avoiding walks on cubic lattice.

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A002902 Number of n-step self-avoiding walks on a cubic lattice with a first step along the positive x, y, or z axis.

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A038515 Number of self-avoiding walks on a tetrahedral (diamond) net, having 2n steps and forming a closed loop.

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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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A010568 Number of 2n-step self-avoiding closed paths on the 4-dimensional cubic lattice.

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A001409 Number of 2n-step polygons on cubic lattice.

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A001667 2n-step polygons on b.c.c. lattice.

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A140476 Number of self-avoiding walks on cubic lattice with no more than n steps.

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A342800 Number of self-avoiding polygons on a 3-dimensional cubic lattice where each walk consists of steps with incrementing length from 1 to n.

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