A336492 Total number of neighbor contacts for n-step self-avoiding walks on a 2D square lattice.

0, 0, 8, 32, 152, 512, 1880, 5920, 19464, 59168, 183776, 545392, 1638400, 4778000, 14043224, 40422544, 116977176, 333346928, 953538440, 2695689520, 7642091352, 21464794032, 60417010152, 168787016352, 472315518008, 1313548558528, 3657850909680, 10133559518800

Offset

1,3

Comments

This sequence gives the total number of neighbor contacts for all n-step self avoiding walks on a 2D square lattices. A neighbor contact is when the walk comes within 1 unit distance of a previously visited point, excluding the previous adjacent point.

Links

Table of n, a(n) for n=1..28.

D. Bennett-Wood, I. G. Enting, D. S. Gaunt, A. J. Guttmann, J. L. Leask, A. L. Owczarek, and S. G. Whittington, Exact enumeration study of free energies of interacting polygons and walks in two dimensions, J. Phys. A: Math. Gen. 31 (1998), 4725-4741.

M. E. Fisher and B. J. Hiley, Configuration and free energy of a polymer molecule with solvent interaction, J. Chem. Phys., 34 (1961), 1253-1267.

A. M. Nemirovsky, K. F. Freed, T. Ishinabe, and J. F. Douglas, Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.

Example

a(1) = a(2) = 0 as a 1 and 2 step walk cannot approach a previous step.
a(3) = 8. The single walk where one interaction occurs, which can be taken in eight ways on a 2D square lattice, is:
.
   +---+
       |
   X---+
.
Therefore, the total number of interactions is 1*1*8 = 8.
a(4) = 32. The four walks where one interaction occurs, each of which can be taken in eight ways on a 2D square lattice, are:
.
  +---+---+   +           +---+       +---+
          |   |               |       |   |
      X---+   +---+   X---+---+   X---+   +
                  |
              X---+
.
Therefore, the total number of interactions is 4*1*8 = 32.
a(5) = 152. Considering only walks which start with one or more steps to the right followed by an upward step there are thirty-five different walks. Eleven of these have one neighbor contact (hence A033155(5) = 11*8 = 88) while four have two contacts. These are:
.
  +---+---+   +---+---+   +---+   +---+
  |       |           |   |       |   |
  +   X---+   X---+---+   +---+   +   +
                              |       |
                          X---+   X---+
.
Therefore, the total number of contacts is (11*1 + 4*2)*8 = 152.

Crossrefs

Cf. A033155 (total number of n-step walks containing one neighbor contact), A038747, A047057, A173380, A174319.

Sequence in context: A253105 A290915 A334693 * A200153 A188121 A045684

Adjacent sequences: A336489 A336490 A336491 * A336493 A336494 A336495

Keyword

nonn

Author

Scott R. Shannon, Jul 23 2020

Status

approved