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

Author

N. J. A. Sloane

Keywords

• nonn
• more

Links

• 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.

Crossrefs

Cf. A010568, A010570.

Programs

Python

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

Extensions

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