A010569 -id:A010569 - cp's OEIS frontend

A010570 Number of 2n-step self-avoiding closed paths on the 6-dimensional cubic lattice.

12, 120, 4200, 216720, 13594320, 959431200, 73286046960, 5928739001280, 501123204523440, 43851618007523760, 3946829550070653840, 363607619806646296800

Offset: 1

N. J. A. Sloane

The initial term a(1) = 12 corresponds to a cycle which is one step in some direction and one step back. Sometimes such paths, in which an edge is used twice, are not counted as self-avoiding path. - M. F. Hasler, Jun 18 2025

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 A8 "Enumeration results for d = 6", 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. A010567 (d=3) - A010569 (d=5).

def A010570(n): # For illustration - becomes slow for n > 4
    if not hasattr(A:=A010570, 'r'):
       A.terms = [12]; A.weights = 12, 120; I = (0,)*6, (1,)+(0,)*5
       A.paths = (*I,(2,)+(0,)*5), (*I,(1,1)+(0,)*4); A.r = tuple(range(6))
    while n > len(A.terms):
        for L in (0, 1):
            np = []; nw=[]; cycles = 0
            for path,weight in zip(A.paths,A.weights):
                end = path[-1]
                for i in A.r:
                   for s in (1, -1):
                      t = tuple(end[j]if j!=i else end[j]+s for j in A.r)
                      if t not in path: np+=[path+(t,)]; nw+=[weight]
                      elif L and t==path[0]: cycles += weight
            A.paths, A.weights = np, nw
        A.terms.append(cycles)
    return A.terms[n-1] # M. F. Hasler, Jun 17 2025

a(6)-a(7) from Sean A. Irvine, Jun 01 2018
a(8) from Sean A. Irvine, Aug 17 2020
"Self-avoiding" added in definition by M. F. Hasler, Jun 18 2025
a(9)-a(12) from Clisby et al.'s data added by Andrei Zabolotskii, Jun 25 2025

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

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A010570 Number of 2n-step self-avoiding closed paths on the 6-dimensional cubic lattice.

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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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