Original entry on oeis.org
1, 0, 8, 4, 48, 68, 284, 684, 1816, 5608, 12684, 42068, 92916, 304100, 688988, 2170020, 5088784, 15436172, 37281880, 109786204, 271062388, 781016892, 1958863988, 5555714820, 14090644980, 39503105472, 101000072900, 280693435596
Offset: 0
N. J. A. Sloane, based on a suggestion of Maurice Craig (Maurice.D.Craig(AT)BHPBilliton.com), May 11 2003
1 + 4*q + 12*q^2 + 36*q^3 + 100*q^4 + 284*q^5 + 780*q^6 + 2172*q^7 + 5916*q^8 + 16268*q^9 + ... / 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + 4*q^8 + 4*q^9 + ... = 1 + 8*q^2 + 4*q^3 + 48*q^4 + 68*q^5 + 284*q^6 + 684*q^7 + 1816*q^8 + 5608*q^9 + ...
A001412
Number of n-step self-avoiding walks on cubic lattice.
Original entry on oeis.org
1, 6, 30, 150, 726, 3534, 16926, 81390, 387966, 1853886, 8809878, 41934150, 198842742, 943974510, 4468911678, 21175146054, 100121875974, 473730252102, 2237723684094, 10576033219614, 49917327838734, 235710090502158, 1111781983442406, 5245988215191414, 24730180885580790, 116618841700433358, 549493796867100942, 2589874864863200574, 12198184788179866902, 57466913094951837030, 270569905525454674614
Offset: 0
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 5.10, pp. 331-339.
- B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 462.
- 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).
- R. D. Schram, G. T. Barkema, and R. H. Bisseling, Table of n, a(n) for n = 0..36
- N. Clisby, Enumerative combinatorics of lattice polymers, Notices AMS, 68:4 (2021), 504-515. (Excellent survey)
- N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. 40 (2007), 10973-11017, Table A5 for n<=30.
- Steven R. Finch, Self-Avoiding-Walk Connective Constants
- 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, J. Phys. A 20 (1987), 1839-1854.
- B. J. Hiley and M. F. Sykes, Probability of initial ring closure in the restricted random-walk model of a macromolecule, J. Chem. Phys., 34 (1961), 1531-1537.
- D. S. McKenzie and C. Domb, The second osmotic virial coefficient of athermal polymer solutions, Proceedings of the Physical Society, 92 (1967) 632-649.
- A. M. Nemirovsky, Karl F. Freed, Takao Ishinabe, and Jack F. Douglas, Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
- D. Randall, Counting in Lattices: Combinatorial Problems from Statistical Mechanics, PhD Thesis (1994).
- Raoul D. Schram, Gerard T. Barkema, and Rob H. Bisseling, Exact enumeration of self-avoiding walks, arXiv:1104.2184 [math-ph], 2011.
- Nobu C. Shirai and Naoyuki Sakumichi, Negative Energetic Elasticity of Lattice Polymer Chain in Solvent, arXiv:2202.12483 [cond-mat.soft], 2022.
- 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, Self-avoiding walks on the simple cubic lattice, J. Chem. Phys., 39 (1963), 410-411.
- M. F. Sykes, A. J. Guttmann, M. G. Watts, and P. D. Roberts, The asymptotic behavior of self-avoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.
- M. F. Sykes, D. S. McKenzie, M. G. Watts, and J. L. Martin, The number of self-avoiding rings on a lattice, J. Phys. A 5 (1972), 661-666.
-
mo = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}}; a[0] = 1;
a[tg_, p_: {{0, 0, 0}}] := Block[{e, mv = Complement[Last[p] + # & /@ mo, p]},
If[tg == 1, Return[Length@mv],Sum[a[tg - 1, Append[p, e]], {e, mv}]]];
a /@ Range[0, 8]
(* Robert FERREOL, Nov 30 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=[[1,0,0],[-1,0,0],[0,1,0],[0,-1,0],[0,0,1],[0,0,-1]]
def a(n,P=[[0,0,0]]):
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(8)]
# Robert FERREOL, Nov 30 2018
A001334
Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.
Original entry on oeis.org
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
- 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.
-
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
A037245
Number of unrooted self-avoiding walks of n steps on square lattice.
Original entry on oeis.org
1, 2, 4, 9, 22, 56, 147, 388, 1047, 2806, 7600, 20437, 55313, 148752, 401629, 1078746, 2905751, 7793632, 20949045, 56112530, 150561752, 402802376, 1079193821, 2884195424, 7717665979, 20607171273, 55082560423, 146961482787, 392462843329, 1046373230168, 2792115083878
Offset: 1
- Bert Dobbelaere, Table of n, a(n) for n = 1..60
- Joerg Arndt, The a(6) = 56 walks of length 6, 2018 (pdf, 2 pages).
- Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.
- Ed Pegg, Jr., Illustrations of polyforms
- Eric Weisstein's World of Mathematics, Polyedge
Asymptotically approaches (1/16) *
A001411.
Cf.
A266549 (closed self-avoiding walks).
A116903
Seaweeds(n): number of n-step self-avoiding walks on upper two quadrants grid starting at origin.
Original entry on oeis.org
1, 3, 7, 19, 49, 131, 339, 899, 2345, 6199, 16225, 42811, 112285, 296051, 777411, 2049025, 5384855, 14190509, 37313977, 98324565, 258654441, 681552747, 1793492411, 4725856129, 12439233695, 32778031159, 86295460555, 227399388019, 598784536563, 1577923781445, 4155176578581
Offset: 0
The 19 seaweeds of length 3. X marks the origin = anchor point.
........................................................
O-O-O...O-O.. .O-O...O-O....O..O-O-O...O......O..O-O....
....|.....|.. .|.....|......|..|.......|......|....|....
....X...X-O....O.....O-X....O..X..O-O..O-O....O....O-X..
...............|............|.......|....|....|.........
O-O.....O-O....X....O.......O..O....O....X..X-O.......O.
|.|.....|...........|.......|..|....|.................|.
X.O...X-O.....O-O...O-O-X...X..O....X.....O.........O-O.
..............|.|..............|..........|.........|...
..X-O-O-O.....O.X...O-O-O-X....O-X....X-O-O.........X...
........................................................
A010575
Number of n-step self-avoiding walks on 4-d cubic lattice.
Original entry on oeis.org
1, 8, 56, 392, 2696, 18584, 127160, 871256, 5946200, 40613816, 276750536, 1886784200, 12843449288, 87456597656, 594876193016, 4047352264616, 27514497698984, 187083712725224, 1271271096363128, 8639846411760440, 58689235680164600, 398715967140863864
Offset: 0
- Hugo Pfoertner, Table of n, a(n) for n = 0..24 [from the Clisby et al. link below]
- N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor., vol. 40 (2007), p. 10973-11017, Table A6 for n <= 24.
- Nathan Clisby, Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps, arXiv:1703.10557 [cond-mat.stat-mech], 30 Mar 2017.
- M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.
- D. MacDonald, D. L. Hunter, K. Kelly, and N. Jan, Self-avoiding walks in two to five dimensions: exact enumerations and series study, J Phys A: Math Gen 25 (1992) Vol. 6, 1429-1440 [Gives 18 terms]
- A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
- Hugo Pfoertner, Results for the 4D Self-Trapping Random Walk
- Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant
a(18) onwards from
R. J. Mathar using data from Clisby et al, Aug 31 2007
A038373
Number of n-step self-avoiding paths on quadrant grid starting at quadrant origin.
Original entry on oeis.org
1, 2, 4, 10, 24, 60, 146, 366, 912, 2302, 5800, 14722, 37368, 95304, 243168, 622518, 1594622, 4094768, 10521384, 27085436, 69768478, 179982688, 464564220, 1200563864, 3104192722, 8034256412, 20803994184, 53915334890, 139785953076, 362681515714, 941361260956, 2444866458524, 6351963691964
Offset: 0
A077482
Number of self-avoiding walks on square lattice trapped after n steps.
Original entry on oeis.org
1, 2, 11, 25, 95, 228, 752, 1860, 5741, 14477, 42939, 109758, 317147, 818229, 2322512, 6030293, 16900541, 44079555, 122379267, 320227677, 882687730, 2315257359, 6346076015, 16675422679, 45502168379, 119728011251, 325510252108, 857400725204
Offset: 7
a(7) = 1 because there is only 1 self-trapping walk with 7 steps: (0,0)(1,0)(1,1)(1,2)(0,2)(-1,2)(-1,1)(0,1); a(8) = 2 because there are 2 self-trapping walks with 8 steps: (0,0)(1,0)(2,0)(2,1)(2,2)(1,2)(0,2)(0,1)(1,1) and (0,0)(1,0)(1,1)(2,1)(3,1)(3,0)(3,-1)(2,-1)(2,0).
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
- 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
A078528
Number of unconstrained walks on square lattice trapped after n steps.
Original entry on oeis.org
1, 1, 2, 5, 15, 30, 76, 170, 422, 961, 2339, 5390, 12977, 30059, 71918, 167019, 397691, 924931, 2194478, 5107991, 12085695, 28143758, 66442935, 154759821, 364706675, 849562628
Offset: 7
a(7)=1 because the unique shortest walk contains no constrained steps. a(10)=5: See illustration in "5 Unconstrained and 7 maximally 2-constrained walks of length 10" given at link.
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