A010568 -id:A010568 - cp's OEIS frontend

A010569 Number of 2n-step self-avoiding closed paths on the 5-dimensional cubic lattice.

10, 80, 2160, 82720, 3737120, 186303840, 9945915840, 558476528000, 32597366872320, 1961752814181280, 121020530395783040, 7620016712806580160

Offset: 1

N. J. A. Sloane

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 A7 "Enumeration results for d = 5", 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. A010568, A010570.

def A010569(n): # For illustration - becomes slow for n >= 5
    if not hasattr(A:=A010569, 'r'):
       A.terms = [10]; A.r = 0,1,2,3,4; z = 0,0,0,0; I = (0,*z), (1,*z)
       A.paths = (*I,(2,*z)), (*I,(1,1,*z[1:])); A.weights = 10, 80
    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(8) from Sean A. Irvine, Jun 04 2018
a(9) from Sean A. Irvine, Aug 10 2020
"Self-avoiding" inserted in definition by M. F. Hasler, Jun 18 2025
a(10)-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

cp's OEIS Frontend

A010569 Number of 2n-step self-avoiding closed paths on the 5-dimensional cubic lattice.

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