login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001411 Number of n-step self-avoiding walks on square lattice.
(Formerly M3448 N1402)
87
1, 4, 12, 36, 100, 284, 780, 2172, 5916, 16268, 44100, 120292, 324932, 881500, 2374444, 6416596, 17245332, 46466676, 124658732, 335116620, 897697164, 2408806028, 6444560484, 17266613812, 46146397316, 123481354908, 329712786220, 881317491628 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
B. Bollobas and O. Riordan, Percolation, Cambridge, 2006, see p. 15.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 5.10, pp. 331-338.
B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 461.
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).
LINKS
N. J. A. Sloane, Hugo Pfoertner, Table of n, a(n) for n = 0..79 (from the Jensen links below)
A. R. Conway et al., Algebraic techniques for enumerating self-avoiding walks on the square lattice, J. Phys A 26 (1993) 1519-1534.
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.
A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.
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.
G. Slade, Self-avoiding walks, Math. Intelligencer, 16 (No. 1, 1994), 29-35.
M. F. Sykes et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.
MAPLE
noloop:=X->evalb(nops(X)=nops({op(X)})):
extend:=proc(L) local L1, U, X, res:
U:=[[1, 0], [0, 1], [-1, 0], [0, -1]]:
res:=NULL:for X in U do L1:=[op(L), L[nops(L)]+X]:
if noloop(L1) then res:=res, L1 fi od:
return(res) end:
walks:={[[0, 0]]}: A001411:=1:
to 12 do walks:=map(x->extend(x), walks): A001411:=A001411, nops(walks) od:
# Robert FERREOL, Mar 29 2019
MATHEMATICA
mo=Tuples[{-1, 1}, 2]; 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, 10] (* Giovanni Resta, May 06 2016 *)
PROG
(Python)
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], [-1, 0], [0, -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, Nov 30 2018; after the Mathematica program.
CROSSREFS
Twice A002900.
Sequence in context: A294782 A002906 A191756 * A095350 A084776 A162921
KEYWORD
nonn,walk,nice
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 07:31 EDT 2024. Contains 370955 sequences. (Running on oeis4.)