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A336492
Total number of neighbor contacts for n-step self-avoiding walks on a 2D square lattice.
2
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
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
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jul 23 2020
STATUS
approved