A078717 -id:A078717 - cp's OEIS frontend

A079156 Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on cubic lattice. Numerator of mean Manhattan displacement s(n)=a(n)/A078717.

10, 67, 396, 2201, 11870, 62571, 324896, 1665349, 8457890, 42605267, 213305636, 1061939193, 5263752278, 25984214383, 127848694424, 627084275649, 3067923454498

Offset: 2

Hugo Pfoertner, Dec 29 2002

A conjectured asymptotic behavior for the mean Manhattan displacement is shown in a diagram lim n-> infinity a(n)/(A078717(n)*n^nu)=c, for some values of nu near 0.59 at Pfoertner link

			a(2)=10 because the A078717(2)=5 different self-avoiding 2-step walks end at (1,0,-1),(1,0,1),(1,-1,0),(1,1,0),(2,0,0)->d=2. a(2)=5*2=10. See also "Distribution of end point distance" at Pfoertner link

See under A001412

Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk
Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant

Cf. A001412, A078717, A078605 (corresponding square displacement).

c Program for distance counting available at Pfoertner link.

a(n)= sum l=1, A078717(n) (|i_l| + |j_l| + |k_l|) where (i_l, j_l, k_l) are the end points of all different self-avoiding n-step walks starting at (0, 0, 0)

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

N. J. A. Sloane

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.

Cf. A002902, A078717, A001411, A001413.

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

A002902 Number of n-step self-avoiding walks on a cubic lattice with a first step along the positive x, y, or z axis.

Original entry on oeis.org

3, 15, 75, 363, 1767, 8463, 40695, 193983, 926943, 4404939, 20967075, 99421371, 471987255, 2234455839, 10587573027, 50060937987, 236865126051, 1118861842047, 5288016609807, 24958663919367, 117855045251079, 555890991721203, 2622994107595707

Offset: 1

N. J. A. Sloane

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).

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 et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
M. F. Sykes, Self-avoiding walks on the simple cubic lattice, J. Chem. Phys., 39 (1963), 410-411.
M. F. Sykes et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.

Equals (1/2)*A001412. Cf. A078717, A001411, A001413.

Name amended by Scott R. Shannon, Sep 17 2020

A078605 Sum of square displacements over all self-avoiding n-step walks on the cubic lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6).

Original entry on oeis.org

1, 12, 97, 672, 4261, 25588, 147821, 830576, 4566917, 24692980, 131682825, 694386864, 3626770709, 18790632772, 96675376705, 494382431552, 2514666026897, 12730690730212, 64177763220925, 322314275563424, 1613192327878789, 8049191357609204, 40048773875769449, 198750753713937600

Offset: 1

Hugo Pfoertner, Dec 09 2002

nonn

A comparison with the conjectured asymptotic behavior of the mean square displacement s(n) over all n-step self-avoiding walks given in Weisstein's article is shown in "Asymptotic Behavior of Mean Square Displacement" at the Pfoertner link.

			a(2)=12 because the A001412(2)/6 = 5 different self-avoiding 2-step walks end at (1,0,-1), (1,0,1), (1,-1,0), (1,1,0)->d^2=2 and at (2,0,0)->d^2=4. a(2) = 4*2 + 1*4 = 12. See also "Distribution of end point distance" at first link.

For references see under A001412

Hugo Pfoertner, Table of n, a(n) for n = 1..36
Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk
Raoul D. Schram, Gerard T. Barkema, and Rob H. Bisseling, Exact enumeration of self-avoiding walks, arXiv:1104.2184 [math-ph], 2011.
Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant.

Cf. A001412, A078717, A079156 (corresponding Manhattan distance sum).

Equals A118313/6.

c Program for distance counting available at Pfoertner link.

a(n) = Sum_{L=1..A001412(n)/6} ( i_L^2 + j_L^2 + k_L^2 ) where (i_L, j_L, k_L) are the endpoints of all different self-avoiding n-step walks.

Terms a(19)-a(36) taken from A118313 by Hugo Pfoertner, Aug 20 2014

Name amended by Scott R. Shannon, Sep 17 2020

A140476 Number of self-avoiding walks on cubic lattice with no more than n steps.

Original entry on oeis.org

1, 7, 37, 187, 913, 4447, 21373, 102763, 490729, 2344615, 11154493, 53088643, 251931385, 1195905895, 5664817573, 26839963627, 126961839601, 600692091703, 2838415775797, 13414448995411, 63331776834145, 299041867336303

Offset: 0

Jonathan Vos Post, Jun 29 2008

nonn

Primes include a(1) = 7, a(2) = 37, a(5) = 4447, a(8) = 102763, a(15) = 26839963627.

			a(9) = 1 + 6 + 30 + 150 + 726 + 3534 + 16926 + 81390 + 387966 + 1853886 = 2344615.

Partial sums of A001412.

Cf. A001411, A001412, A001413, A002902, A078717.

cp's OEIS Frontend

A079156 Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on cubic lattice. Numerator of mean Manhattan displacement s(n)=a(n)/A078717.

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A001412 Number of n-step self-avoiding walks on cubic lattice.

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A002902 Number of n-step self-avoiding walks on a cubic lattice with a first step along the positive x, y, or z axis.

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A078605 Sum of square displacements over all self-avoiding n-step walks on the cubic lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6).

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A140476 Number of self-avoiding walks on cubic lattice with no more than n steps.

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