A001336 -id:A001336 - 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

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

A001666 Number of n-step self-avoiding walks on b.c.c. lattice (version 2).

Original entry on oeis.org

1, 8, 56, 392, 2648, 17960, 120056, 804824, 5351720, 35652680, 236291096, 1568049560, 10368669992, 68626647608, 453032542040, 2992783648424, 19731335857592, 130161040083608, 857282278813256, 5648892048530888, 37175039569217672, 244738250638121768

Offset: 0

N. J. A. Sloane

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..28 (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.
A. J. Guttmann, On the critical behavior of self-avoiding walks II, J. Phys. A 22 (1989), 2807-2813. See Table 1.
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 I.
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.
M. F. Sykes et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.
Index entries for sequences related to b.c.c. lattice

Equals twice A002903 except for initial term.

Cf. A001336, A001412.

a(16)-a(18) from Bert Dobbelaere, Jan 16 2019
Terms a(19) and beyond from Schram et al. added by Andrey Zabolotskiy, Feb 02 2022
Edited by N. J. A. Sloane, Oct 16 2022

A227511 Triangle read by rows: Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = k.

Original entry on oeis.org

1, 4, 4, 36, 32, 16, 308, 292, 192, 64, 2764, 2672, 2016, 1024, 256, 25404, 24780, 20160, 12480, 5120, 1024, 237164, 232512, 197940, 137472, 71680, 24576, 4096, 2237948, 2201948, 1930944, 1443616, 869376, 390144, 114688, 16384, 21286548, 20997008, 18805488, 14786176

Offset: 0

Joseph Myers, Jul 14 2013

The number of walks ending with x = -k is the same as the number ending with x = k.

			Initial rows (paths of length 0, 1, 2, ...):
{ 1 };
{ 4, 4 };
{ 36, 32, 16 };
{ 308, 292, 192, 64 }.

Bert Dobbelaere, Table of n, a(n) for n = 0..135 (terms 0..77 from Joseph Myers)
J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.

Cf. A000765, A000766, A000767, A000768, A001336.

A000765 Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 0.

Original entry on oeis.org

1, 4, 36, 308, 2764, 25404, 237164, 2237948, 21286548, 203701772, 1958748676, 18908954324, 183135542956, 1778581016076, 17314029758828, 168891863875652

Offset: 0

N. J. A. Sloane

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).

J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.
Index entries for sequences related to f.c.c. lattice

Cf. A000766, A000767, A000768, A001336, A227511.

Edited and extended by Joseph Myers, Jul 14 2013

a(12)-a(15) from Bert Dobbelaere, Jan 13 2019

A000766 Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 1.

Original entry on oeis.org

4, 32, 292, 2672, 24780, 232512, 2201948, 20997008, 201314448, 1938659936, 18737136032, 181646110192, 1765522809468, 17198432462368, 167859941774728

Offset: 1

N. J. A. Sloane

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).

J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.
Index entries for sequences related to f.c.c. lattice

Cf. A000765, A000767, A000768, A001336, A227511.

Edited and extended by Joseph Myers, Jul 14 2013

a(12)-a(15) from Bert Dobbelaere, Jan 13 2019

A000767 Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 2.

Original entry on oeis.org

16, 192, 2016, 20160, 197940, 1930944, 18805488, 183156320, 1785303660, 17421627280, 170214459928, 1665089608504, 16307758577692, 159896665015064

Offset: 2

N. J. A. Sloane

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).

J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.
Index entries for sequences related to f.c.c. lattice

Cf. A000765, A000766, A000768, A001336, A227511.

Edited and extended by Joseph Myers, Jul 14 2013

a(12)-a(15) from Bert Dobbelaere, Jan 13 2019

A000768 Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 3.

Original entry on oeis.org

64, 1024, 12480, 137472, 1443616, 14786176, 149371964, 1496777088, 14924375156, 148353336272, 1471838989872, 14584911842000, 144423054478680

Offset: 3

N. J. A. Sloane

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).

J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.
Index entries for sequences related to f.c.c. lattice

Cf. A000765, A000766, A000767, A001336, A227511.

Edited and extended by Joseph Myers, Jul 14 2013

a(12)-a(15) from Bert Dobbelaere, Jan 13 2019

A336907 Number of n-step self-avoiding walks on the f.c.c. lattice with no non-contiguous adjacencies.

Original entry on oeis.org

1, 12, 84, 564, 3804, 25308, 167796, 1108452, 7305132, 48043044, 315466884, 2068604028, 13549151244, 88658702484, 579649775796, 3786953782764

Offset: 0

Sean A. Irvine, Aug 07 2020

Cf. A001336, A174319 (cubic lattice equivalent), A336906 (b.c.c. equivalent).

A213453 Self-avoiding walks on the f.c.c. lattice.

Original entry on oeis.org

1, 6, 66, 702, 7350, 76266, 786858, 8086074, 82848522, 846886962, 8640964782

Offset: 1

N. J. A. Sloane, Jun 12 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

D. S. McKenzie and C. Domb, The second osmotic virial coefficient of athermal polymer solutions, Proceedings of the Physical Society, 92 (1967) 632-649.

Cf. A001336, A002902.

Apparently a(n) = A001336(n-1)/2 for n > 1. - Andrey Zabolotskiy, Dec 29 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A001666 Number of n-step self-avoiding walks on b.c.c. lattice (version 2).

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A227511 Triangle read by rows: Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A000765 Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 0.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A000766 Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 1.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A000767 Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 2.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A000768 Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 3.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A336907 Number of n-step self-avoiding walks on the f.c.c. lattice with no non-contiguous adjacencies.

Views

Author

Keywords

Crossrefs

A213453 Self-avoiding walks on the f.c.c. lattice.

Views

Author

Keywords

References

Links