A036418 -id:A036418 - cp's OEIS frontend

A001334 Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.

1, 6, 30, 138, 618, 2730, 11946, 51882, 224130, 964134, 4133166, 17668938, 75355206, 320734686, 1362791250, 5781765582, 24497330322, 103673967882, 438296739594, 1851231376374, 7812439620678, 32944292555934, 138825972053046

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-234 of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989.
B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 459.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Jensen, Table of n, a(n) for n = 0..40 (from the Jensen link below)
Sergio Caracciolo, Anthony J. Guttmann, Iwan Jensen, Andrea Pelissetto, Andrew N. Rogers, Alan D. Sokal, Correction-to-Scaling Exponents for Two-Dimensional Self-Avoiding Walks, Journal of Statistical Physics, September 2005, Volume 120, Issue 5, pp. 1037-1100.
M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-13, 56-57, 142-143, 150-151 from of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989. (Annotated scanned copy)
A. J. Guttmann and J. Wang, The extension of self-avoiding random walk series in two dimensions, J. Phys. A 24 (1991), 3107-3109.
B. D. Hughes, Random Walks and Random Environments, vol. 1, Oxford 1995, Tables and references for self-avoiding walks counts [Annotated scanned copy of several pages of a preprint or a draft of chapter 7 "The self-avoiding walk"]
I. Jensen, Series Expansions for Self-Avoiding Walks
J. L. Martin, M. F. Sykes and F. T. Hioe, Probability of initial ring closure for self-avoiding walks on the face-centered cubic and triangular lattices, J. Chem. Phys., 46 (1967), 3478-3481.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
D. C. Rapaport, End-to-end distance of linear polymers in two dimensions: a reassessment, J. Phys. A 18 (1985), L201.
S. Redner, Distribution functions in the interior of polymer chains, J. Phys. A 13 (1980), 3525-3541, doi:10.1088/0305-4470/13/11/023.
M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62.
Joris van der Hoeven, On asymptotic extrapolation, Journal of symbolic computation, 2009, p. 1010.

Cf. A001411, A001412, A001336, A001666, A001335, A036418, A192208.

mo={{2, 0},{-1, 1},{-1, -1},{-2, 0},{1, -1},{1, 1}}; a[0]=1;
a[tg_, p_:{{0, 0}}] := Block[{e, mv = Complement[Last[p]+# & /@ mo, p]}, If[tg == 1, Length@mv, Sum[a[tg-1, Append[p, e]], {e, mv}]]];
a /@ Range[0, 6]
(* Robert FERREOL, Nov 28 2018; after the program of Giovanni Resta in A001411 *)

def add(L,x):
    M=[y for y in L];M.append(x)
    return(M)
plus=lambda L,M : [x+y for x,y in zip(L,M)]
mo=[[2,0],[-1,1],[-1, -1],[-2,0],[1,-1],[1, 1]]
def a(n,P=[[0, 0]]):
    if n==0: return(1)
    mv1 = [plus(P[-1],x) for x in mo]
    mv2=[x for x in mv1 if x not in P]
    if n==1: return(len(mv2))
    else: return(sum(a(n-1,add(P,x)) for x in mv2))
[a(n) for n in range(11)]
# Robert FERREOL, Dec 11 2018

A284869 Number of n-step 2-dimensional closed self-avoiding paths on triangular lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 5, 16, 37, 120, 344, 1175, 3807, 13224, 45645, 161705, 575325, 2074088, 7521818, 27502445, 101134999, 374128188

Offset: 1

Luca Petrone, Apr 04 2017

Differs from A057729 beginning at n = 11, since that sequence includes triangular polyominoes with holes.

a(n) is the number of simply connected polyiamonds with perimeter n. - Walter Trump, Nov 29 2023

Rade Doroslovački, Ivan Stojmenović and Ratko Tošić, Generating and counting triangular systems, BIT Numerical Mathematics, 27 (1987), 18-24. See Table 1.
Hugo Pfoertner, Illustration of ratio A036418(n)/a(n) using Plot2.
Hugo Pfoertner, Illustration of polygons of perimeter <= 11.
Walter Trump, Self-avoiding closed walks on a triangular lattice

Approaches (1/12)*A036418 for increasing n.

Cf. A057729, A258206, A266549, A316196.

a(15) from Hugo Pfoertner, Jun 27 2018

a(16)-a(22) from Walter Trump, Nov 29 2023

A316196 Number of symmetric self-avoiding polygons on hexagonal lattice with perimeter n, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 3, 10, 9, 36, 27, 129, 90, 449, 331

Offset: 1

Hugo Pfoertner, Jun 27 2018

Hugo Pfoertner, Illustration of polygons of perimeter <= 11.

Cf. A036418, A284869, A306176.

A337550 Number of closed-loop self-avoiding paths of length 4n on a 2D square lattice where no step can be in the same direction as the previous step.

Original entry on oeis.org

8, 0, 24, 64, 360, 1728, 8624, 43776, 225216, 1173280, 6182704, 32905536, 176657000, 955629920, 5204178360, 28509374976, 157005901896, 868756900608, 4827586102216, 26929911745600, 150750954809952, 846588050093632, 4768197762850608

Offset: 1

Scott R. Shannon, Aug 31 2020

See A337353 for the corresponding number of walks.
Only walks with a length of 4n (except for n=2) can create closed loops.
From Pontus von Brömssen, May 06 2025: (Start)
A006782 counts the walks up to starting point and direction of the walk.
A156228 counts the walks up to rotations, reflections, starting point, and direction of the walk.
(End)

			a(1) = 8. The single walk of length 4 is:
.
+---+
|   |
+---+
.
This can be taken in 8 different ways on a square lattice, giving a total 1*8 = 8.
a(2) = 0 as there is no closed-loop path consisting of 8 steps.
a(3) = 24. There is one walk, ignoring reflection and rotations, with a length of 12. The walk is:
.
    +---+
    |   |
+---+   +---+
|           |
+---+   +---+
    |   |
    +---+
.
This can be walked in 3 different ways if the first steps are right and then upward. This path can be then taken in 8 ways on a square lattice, giving a total number of 3*8 = 24.
a(4) = 64. There is one walk, with indistinct reflections and rotations, with a length of 16. The walk is:
.
        +---+
        |   |
    +---+   +---+
    |           |
+---+       +---+
|           |
+---+   +---+
    |   |
    +---+
.
This can be walked in 8 different ways if the first steps are right and then upward. This path can be then taken in 8 ways on a square lattice, giving a total number of 8*8 = 64.
.
a(5) = 360. There are four walks, with indistinct reflections and rotations, with a length of 20. The walks, and the different ways they can be taken, are:
.
            +---+              +---+
            |   |              |   |
        +---+   +---+      +---+   +---+
        |           |      |           |
    +---+       +---+      +---+       +---+
    |           |              |           |
+---+       +---+          +---+       +---+
|           |              |           |
+---+   +---+              +---+   +---+
    |   |     x 10             |   |     x 20
    +---+                      +---+
        +---+                  +---+
        |   |                  |   |
    +---+   +---+          +---+   +---+
    |           |          |           |
+---+           +---+      +---+   +---+
|                   |          |   |
+---+           +---+      +---+   +---+
    |           |          |           |
    +---+   +---+          +---+   +---+
        |   |    x 5           |   |     x 10
        +---+                  +---+
.
Each of these can be walked in 8 different ways on a square lattice, giving a total number of 8*(10+20+5+10) = 360.
See the attached text file for images of the closed-loops for n=1 to n=11.

A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.
Scott R. Shannon, Text images of the closed-loops for n=1 to n=11.

Cf. A337353, A010566, A334720, A036418.

Cf. A006782, A156228.

a(n) = 8*n*A006782(n). - Pontus von Brömssen, May 06 2025

a(18)-a(19) from Bert Dobbelaere, Sep 09 2020

a(20)-a(23) (using A006782 data) from Pontus von Brömssen, May 06 2025

A341630 Number of fixed polyiamonds of area n without holes.

Original entry on oeis.org

2, 3, 6, 14, 36, 94, 250, 675, 1832, 5005, 13746, 37901, 104902, 291312, 811346, 2265905, 6343854, 17801383, 50057400, 141034248, 398070362, 1125426581, 3186725646, 9036406687, 25658313188, 72946289247, 207628101578, 591622990214, 1687527542874, 4818113792640

Offset: 1

Andrey Zabolotskiy, Feb 16 2021

nonn

Equivalently, closed self-avoiding paths on the hexagonal net, where rotations and reflections of the whole path are not allowed and there is no selected starting point, with enclosed area n.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..60 (from Iwan Jensen's table)
Anthony J. Guttmann, editor, Polygons, Polyominoes and Polycubes, LNP 775, Springer, 2009. See Table 16.8 "Triangular lattice SAP by area" on p. 476 with reference "Iwan Jensen, Unpublished".
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A001420 (polyiamonds with holes allowed; first deviates at n=9), A036418 (polyiamonds with given perimeter, i.e. paths with given length), A070765 (free polyiamonds, i.e. reduced for symmetry: rotations and reflections are allowed), A006724 (analog for square lattice).

A387209 Number of convex polygons with perimeter n on the regular triangular lattice, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 4, 6, 5, 10, 7, 12, 11, 16, 13, 22, 17, 26, 23, 32, 27, 41, 33, 47, 42, 56, 48, 68, 57, 77, 69, 89, 78, 105, 90, 117, 106, 133, 118, 153, 134, 169, 154, 189, 170, 214, 190, 234, 215, 259, 235, 289, 260, 314, 290, 344, 315, 380

Offset: 0

Walter Trump, Aug 22 2025

a(n) also is the number of convex polyiamonds (triangular polyominoes) with perimeter n.

Walter Trump, Table of n, a(n) for n = 0..10000
Walter Trump, Convex Polyiamonds
Walter Trump, Examples for a(3)-a(12)

Cf. A096004 (number of convex polyiamonds with n cells), A284869 (including nonconvex but simply connected polyiamonds with perimeter n), A057729 (including polyiamonds with holes), A036418 (including rotations and reflections but no holes).

cp's OEIS Frontend

A001334 Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.

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A284869 Number of n-step 2-dimensional closed self-avoiding paths on triangular lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

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A316196 Number of symmetric self-avoiding polygons on hexagonal lattice with perimeter n, not counting rotations and reflections as distinct.

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A337550 Number of closed-loop self-avoiding paths of length 4n on a 2D square lattice where no step can be in the same direction as the previous step.

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A341630 Number of fixed polyiamonds of area n without holes.

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A387209 Number of convex polygons with perimeter n on the regular triangular lattice, not counting rotations and reflections as distinct.

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