A010567 2n-step 3-dimensional closed paths on cubic lattice.

6, 24, 264, 3312, 48240, 762096, 12673920, 218904768, 3891176352, 70742410800, 1309643747808, 24609869536800, 468270744898944, 9005391024862848, 174776445357365040, 3419171337633496704

Offset

1,1

Comments

This sequence counts self-avoiding closed paths and 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

Links

Table of n, a(n) for n=1..16.

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

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.

Sequence in context: A052671 A052733 A323449 * A097171 A152886 A128614

Adjacent sequences: A010564 A010565 A010566 * A010568 A010569 A010570

Keyword

nonn,more

Author

N. J. A. Sloane

Extensions

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

Status

approved