A001334 -id:A001334 - cp's OEIS frontend

A177414 Partial sums of A001334.

1, 7, 37, 175, 793, 3523, 15469, 67351, 291481, 1255615, 5388781, 23057719, 98412925, 419147611, 1781938861, 7563704443, 32061034765, 135735002647, 574031742241, 2425263118615, 10237702739293, 43181995295227, 182007967348273

Offset: 0

Jonathan Vos Post, Dec 10 2010

Number of self-avoiding walks of no more than n-steps on hexagonal [ =triangular ] lattice.

Cf. A001334.

a(n) = Sum_{i=0..n} A001334(i).

a(19) corrected by Georg Fischer, Aug 29 2020

A151541 Number of 2-sided triangular strip polyedges with n cells.

Original entry on oeis.org

1, 3, 8, 32, 123, 523, 2201, 9443, 40341, 172649, 736926, 3141607, 13367012, 56790498, 240919918, 1020753475, 4319803799, 18262494912, 77134873774, 325518862387, 1372679840360, 5784417772262

Offset: 1

Ed Pegg Jr, May 13 2009

Also number of unrooted self-avoiding walks of n steps on hexagonal [ =triangular ] lattice. - Hugo Pfoertner, Jun 23 2018

Ed Pegg, Jr., Illustrations of polyforms
Hugo Pfoertner, Illustration of ratio A001334(n)/a(n) using Plot2.
Eric Weisstein's World of Mathematics, Polyedge

Cf. A037245, A151539, A151540, A159867, A266925.

Asymptotically approaches (1/24)*A001334(n) for increasing n.

a(9)-a(13) from Joseph Myers, Oct 05 2011

a(14)-a(22) from Bert Dobbelaere, Mar 23 2025

A001336 Number of n-step self-avoiding walks on f.c.c. lattice.

Original entry on oeis.org

1, 12, 132, 1404, 14700, 152532, 1573716, 16172148, 165697044, 1693773924, 17281929564, 176064704412, 1791455071068, 18208650297396, 184907370618612, 1876240018679868, 19024942249966812, 192794447005403916, 1952681556794601732, 19767824914170222996

Offset: 0

N. J. A. Sloane

B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 460.
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).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..24 (from Schram et al.)
M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
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"]
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.
S. McKenzie, Self-avoiding walks on the face-centered cubic lattice, J. Phys. A 12 (1979), L267-L270.
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.
Raoul D. Schram, Gerard T. Barkema, Rob H. Bisseling and Nathan Clisby, Exact enumeration of self-avoiding walks on BCC and FCC lattices, J. Stat. Mech. (2017) 083208; arXiv:1703.09340 [cond-mat.stat-mech], 2017. See Table II.
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.
Index entries for sequences related to f.c.c. lattice

Cf. A001411, A001412, A001334, A001666, A001337.

a(15) from Bert Dobbelaere, Jan 13 2019

Terms a(16) and beyond from Schram et al. added by Andrey Zabolotskiy, Feb 02 2022

A001335 Number of n-step polygons on hexagonal lattice.

Original entry on oeis.org

1, 0, 0, 12, 24, 60, 180, 588, 1968, 6840, 24240, 87252, 318360, 1173744, 4366740, 16370700, 61780320, 234505140, 894692736, 3429028116, 13195862760, 50968206912, 197517813636, 767766750564, 2992650987408, 11694675166500, 45807740881032

Offset: 0

N. J. A. Sloane

A "polygon" is a self-avoiding walk from (0,0) to (0,0).

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, personal communication.
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).

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, On Two-Dimensional Self-Avoiding Random Walks, J. Phys. A 17 (1984), 455-468.
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"]
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
M. F. Sykes et al., The number of self-avoiding walks on a lattice, J. Phys. A 5 (1972), 661-666.

Equals 6*A003289(n-1), n > 2.

Cf. A001334.

a(22)-a(25) computed from A003289 by Bert Dobbelaere, Jan 04 2019

a(26) from Bert Dobbelaere, Jan 15 2019

A001337 Number of n-step polygons on f.c.c. lattice.

Original entry on oeis.org

0, 0, 48, 264, 1680, 11640, 86352, 673104, 5424768, 44828400, 377810928, 3235366752, 28074857616, 246353214240, 2182457514960, 19495053028800, 175405981214592

Offset: 1

N. J. A. Sloane

Number of n-step closed self-avoiding walks starting at the origin. - Bert Dobbelaere, Jan 14 2019

B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 460.
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).

M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
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"]
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.
Index entries for sequences related to f.c.c. lattice

Equals 12*A003287(n-1), n > 1.
Equals 2n*A005398(n).
Cf. A001336.

a(15)-a(17) from Bert Dobbelaere, Jan 14 2019

A036418 Number of self-avoiding polygons with perimeter n on hexagonal [ =triangular ] lattice.

Original entry on oeis.org

0, 0, 2, 3, 6, 15, 42, 123, 380, 1212, 3966, 13265, 45144, 155955, 545690, 1930635, 6897210, 24852576, 90237582, 329896569, 1213528736, 4489041219, 16690581534, 62346895571, 233893503330, 880918093866, 3329949535934

Offset: 1

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.

B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 459.

I. Jensen, Table of n, a(n) for n = 1..60
Iwan Jensen, Self-avoiding walks and polygons on the triangular lattice, arXiv:cond-mat/0409039 [cond-mat.stat-mech], 2004.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A001334, A284869.

A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi(2k/n - 1), k = 1..n-1.

Original entry on oeis.org

71, 71, 40, 77, 45, 51, 42, 56, 49, 51, 48, 54

Offset: 3

Hugo Pfoertner, Dec 27 2018

The cases n = 3, 4, and 6 correspond to the usual self-avoiding random walks on the honeycomb net, the square lattice, and the hexagonal lattice, respectively. The other cases n = 5, 7, ... are a generalization using self-avoiding rooted walks similar to those defined in A306175, A306177, ..., A306182. The walk is trapped if it cannot be continued without either hitting an already visited (lattice) point or crossing or touching any straight line connecting successively visited points on the path up to the current point.
The result 71 for n=4 was established in 1984 by Hemmer & Hemmer.
The sequence data are based on the following results of at least 10^9 simulated random walks for each n <= 12, with an uncertainty of +- 0.004 for the average walk length:
   n  length
71.132
70.760 (+-0.001)
40.375
77.150
45.297
51.150
42.049
56.189
48.523
51.486
47.9   (+-0.2)
53.9   (+-0.2)

S. Hemmer, P. C. Hemmer, An average self-avoiding random walk on the square lattice lasts 71 steps, J. Chem. Phys. 81, 584 (1984)
Hugo Pfoertner, Examples of self-trapping random walks.
Hugo Pfoertner, Probability density for the number of steps before trapping occurs, 2018.
Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk.
Alexander Renner, Self avoiding walks and lattice polymers, Diplomarbeit, Universität Wien, December 1994.

Cf. A001668, A001411, A001334, A077482, A306175, A306177, A306178, A306179, A306180, A306181, A306182.

Cf. A122223, A122224, A122226, A127399, A127400, A127401, A300665, A323141, A323560, A323562, A323699.

A249795 Self-avoiding walks with n steps on the truncated trihexagonal tiling or (4,6,12) lattice.

Original entry on oeis.org

1, 3, 6, 12, 22, 42, 78, 146, 264, 490, 894, 1646, 3012, 5528, 10086, 18476, 33648, 61472, 111702, 203552, 368872, 670538, 1213118, 2201208, 3980380, 7214200, 13044916, 23627064, 42714902, 77316682, 139695536, 252664214, 456138008, 824332804, 1487051098, 2685425808

Offset: 0

Mike Zabrocki, Nov 05 2014

A self-avoiding walk is a sequence of adjacent points in a lattice that are all distinct.
The truncated trihexagonal tiling or (4,6,12) lattice is one of eight semi-regular tilings of the plane. Each vertex of the lattice is adjacent to a square, hexagon and a 12-sided polygon with sides of equal length.
It is also the Cayley graph of the affine G2 Coxeter group generated by three generators {s_0, s_1, s_2} with the relations (s_0 s_2)^2 = (s_0 s_1)^3 = (s_1 s_2)^6 = 1.

			There are 6 paths of length 2 on the (4,6,12) lattice corresponding to the reduced words in the Coxeter group s_0 s_2, s_0 s_1, s_1 s_2, s_1 s_0, s_2 s_0, s_2 s_1.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..47 (from Alm, 2005; terms 0..42 from Sean A. Irvine)
Sven Erick Alm, Upper and lower bounds for the connective constants of self-avoiding walks on the Archimedean and Laves lattices, J. Phys. A.: Math. Gen., 38 (2005), 2055-2080. Also technical report of the same name, 2004. See Table 2, column (4.6.12).
Sean A. Irvine, Java program (github)
Wikipedia, truncated trihexagonal tiling

Cf. A001411 (square lattice), A001334 (hexagonal lattice), A249565 (truncated square tiling), A326743 (dual, degree 12 vertex), A326744 (dual, degree 6 vertex), A326745 (dual, degree 4 vertex).

a(15)-a(19) corrected by Mike Zabrocki and Sean A. Irvine, Jul 25 2019

More terms from Sean A. Irvine, Jul 25 2019

A306176 Number of symmetric unrooted self-avoiding walks of n steps on hexagonal lattice.

Original entry on oeis.org

1, 3, 4, 12, 18, 50, 78, 208, 337, 867, 1440, 3613, 6133, 15058, 26037, 62756, 110274, 261524, 466183, 1089717, 1967725, 4540103, 8294946, 18913123, 34929064, 78778601, 146943175, 328097471, 617667052, 1366311508, 2594446392, 5689203511, 10890701272, 23687039917, 45689624111, 98612105026

Offset: 1

Hugo Pfoertner, Jun 25 2018

Hugo Pfoertner, Illustration of walks of lengths <= 7.

Cf. A001334, A151541.

a(10)-a(36) from Bert Dobbelaere, Mar 23 2025

A323132 Number of uncrossed unrooted knight's paths of length n on an infinite board.

Original entry on oeis.org

1, 6, 25, 160, 966, 6018, 37079, 227357

Offset: 1

Hugo Pfoertner, Jan 05 2019

Paths which are equivalent under rotation, reflection or reversal are counted only once.

			See illustrations at Pfoertner link.

Hugo Pfoertner, Illustrations of unrooted uncrossed knight's paths of length <= 4, (2019).

Cf. A003192, A001334, A151541, A272773, A306178, A323131, A323133, A323134.

cp's OEIS Frontend

A177414 Partial sums of A001334.

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

A151541 Number of 2-sided triangular strip polyedges with n cells.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A001336 Number of n-step self-avoiding walks on f.c.c. lattice.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A001335 Number of n-step polygons on hexagonal lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A001337 Number of n-step polygons on f.c.c. lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A036418 Number of self-avoiding polygons with perimeter n on hexagonal [ =triangular ] lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi*(2*k/n - 1), k = 1..n-1.

Views

Author

Keywords

Comments

Links

Crossrefs

A249795 Self-avoiding walks with n steps on the truncated trihexagonal tiling or (4,6,12) lattice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A306176 Number of symmetric unrooted self-avoiding walks of n steps on hexagonal lattice.

Views

Author

Keywords

Links

Crossrefs

Extensions

A323132 Number of uncrossed unrooted knight's paths of length n on an infinite board.

Views

Author

Keywords

Comments

Examples

A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi(2k/n - 1), k = 1..n-1.