A001412 -id:A001412 - cp's OEIS frontend

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).

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

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

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

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

N. J. A. Sloane

The computation for n=16 took 11.5 days CPU time on a 500MHz Digital Alphastation. The asymptotic behavior lim n->infinity a(n)/mu^n=const is discussed in the MathWorld link. The Pfoertner link provides an illustration of the asymptotic behavior indicating that the connective constant mu is in the range [6.79,6.80]. - Hugo Pfoertner, Dec 14 2002

Computation of the new term a(17) took 16.5 days CPU time on a 1.5GHz Intel Itanium 2 processor. - Hugo Pfoertner, Oct 19 2004

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

Cf. A001411, A001412, A242355, A323856, A323857, A366925.

c A "brute force" Fortran program to count the 4D walks is available at the Pfoertner link.

a(n) = 8*A366925(n) for n >= 1. - Hugo Pfoertner, Nov 03 2023

a(12)-a(16) from Hugo Pfoertner, Dec 14 2002
a(17) from Hugo Pfoertner, Oct 19 2004
a(18) onwards from R. J. Mathar using data from Clisby et al, Aug 31 2007

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

A116904 Number of n-step self-avoiding walks on the upper 4 octants of the cubic grid starting at origin.

Original entry on oeis.org

1, 5, 21, 93, 409, 1853, 8333, 37965, 172265, 787557, 3593465, 16477845, 75481105, 346960613, 1593924045, 7341070889, 33798930541, 155915787353, 719101961769, 3321659652529, 15341586477457, 70944927549085, 328054694768261, 1518490945278377, 7028570356547189, 32560476643826933, 150838831585499069

Offset: 0

Giovanni Resta, Feb 15 2006

nonn

Guttmann-Torrie simple cubic lattice series coefficients c_n^{2}(Pi). - N. J. A. Sloane, Jul 06 2015

			See A116903 for a graphical example of the bidimensional counterpart.

M. N. Barber et al., Some tests of scaling theory for a self-avoiding walk attached to a surface, 1978 J. Phys. A: Math. Gen. 11 1833.
Nathan Clisby, Andrew R. Conway and Anthony J. Guttmann, Three-dimensional terminally attached self-avoiding walks and bridges, J. Phys. A: Math. Theor., 49 (2016), 015004; arXiv:1504.02085 [cond-mat.stat-mech], 2015. [Warning: arXiv version has typos in a(11) and a(12).]
T. Dachraoui et al., Elementary paths in a cubic lattice and application to molecular biology, Kybernetes, Vol. 26 No. 9, pp. 1012-1030.
A. J. Guttmann and G. M. Torrie, Critical behavior at an edge for the SAW and Ising model, J. Phys. A 17 (1984), 3539-3552.

Cf. A001412, A039648, A116903.

a(16)-a(20) from Scott R. Shannon, Aug 12 2020

a(21)-a(26) from Clisby et al. added by Andrey Zabolotskiy, Apr 18 2023

A337023 Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2h X 2h X 2h where the walk starts at the center of the box.

Original entry on oeis.org

6, 24, 6, 72, 30, 6, 168, 144, 30, 6, 456, 624, 150, 30, 6, 1032, 2520, 720, 150, 30, 6, 2712, 9360, 3408, 726, 150, 30, 6, 5784, 34008, 15432, 3528, 726, 150, 30, 6, 14640, 120960, 68088, 16776, 3534, 726, 150, 30, 6, 29760, 430656, 289128, 79320, 16920, 3534, 726, 150, 30, 6

Offset: 1

Scott R. Shannon, Aug 11 2020

			T(1,2) = 24 as after a step in one of the six axial directions the walk must turn along the face of the box; this eliminates the 2-step straight walk in all directions, so the total number of walks is 6*5-6 = 24.
.
The table begins:
.
6 24  72 168  456  1032  2712   5784   14640   29760    71136    133344    291696..
6 30 144 624 2520  9360 34008 120960  430656 1511856  5340312  18587208  65176416..
6 30 150 720 3408 15432 68088 289128 1205976 4920528 19985928  80066136 321160728..
6 30 150 726 3528 16776 79320 366960 1677864 7516992 33312456 145379760 630249720..
6 30 150 726 3534 16920 81216 385224 1822584 8518920 39588480 181800312 829567656..
6 30 150 726 3534 16926 81384 387768 1850376 8765304 41478144 194837136 912538512..
6 30 150 726 3534 16926 81390 387960 1853664 8805504 41872944 198158520 937459176..
6 30 150 726 3534 16926 81390 387966 1853880 8809632 41928816 198761160 942984312..
6 30 150 726 3534 16926 81390 387966 1853886 8809872 41933880 198836352 943868424..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934144 198842448 943966968..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842736 943974192..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974504..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974510..

Cf. A001412 (h->infinity), A337021 (h=1), A337031 (start at center of face), A337035 (start as corner of box), A336862 (start at middle of edge), A116904, A039648.

For n <= h, T(h,n) = A001412(n).
Row 1 = T(1,n) = A337021(n).
For n >= (2h+1)^3, T(h,n) = 0 as the walk contains more steps than there are available lattice points in the 2h X 2h X 2h box.

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

A335780 The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where both the nodes and connecting rods have mass.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 7, 15, 37, 65, 115, 223, 503, 1127, 2761, 6225, 15393, 34915, 84399, 193489, 477727, 1113059, 2753799, 6486011, 16181965, 38447093, 95995579

Offset: 1

Scott R. Shannon, Sep 13 2020

Consider a self-avoiding walk on a 2D square lattice where each visited node is given a fixed mass and each node is connected by a rod of another fixed mass. Hang the resulting lattice structure from a string at the first node. This sequence gives the number of walks of length n such that the structure will hang perfectly vertically, and will return to this position if perturbed.
For a walk to be stable requires the torque around the first node to be zero for both the node and rod masses, and that the overall center of mass of the structure is lower than the first node. As n increases the number of walks satisfying these conditions decreases rapidly. For example the total number of 2D self-avoiding walks on a square lattice in the lower two quadrants for n=27 is A116903(27) = 227399388019. The total number of hanging stable walks for n=27 is 95995579, indicating only one in about 2370 walks is stable.
For all stable walks it is found that the final node is always directly underneath the starting node. This is not the case if only the node or rod masses are considered.
See A337761 for the equalivalent sequence on a 3D cubic lattice.

			a(1)-a(5) = 1 as the only stable walk is a walk straight down from the first node.
a(6) = 3. There is one stable walk with a first step to the right:
.
      X-----+
            |
            |
+-----+-----+
|
|
+-----+
.
where 'X' represents the hanging point first node at (0,0).
Assuming a mass of p for the nodes, q for the rods, and a length l for the rods, the total torque from the nodes to the right of the first node is 2*p*l, which equals that from the nodes to the left. The total torque for the rods to the right of the first node is 2*q*(1/2)*l + 1*q*1*l = 2ql, which equals that from the rods to the left. The center of mass is at coordinate (0,-1). This walk can be taken in 2 ways thus, with the straight down walk, the total number of stable walks is 2+1 = 3.
a(20) = 193489. An example of a 20-step stable walk is:
.
            X---+
                |
    +---+       +---+---+
    |   |               |
    +   +---+---+       +
    |           |       |
+---+           +---+---+
|
+---+---+---+
.
The total torque from the nodes to the right of the first node is 4*p*1*l + 2*p*2*l + 3*p*3*l = 17pl. The torque from the left nodes is 3*p*1*l + 4*p*2*l + 2*p*3*l = 17pl. The total torque from the rods to the right of the first node is 2*q*(l/2)*l + 2*q*1*l + 2*q*(3/2)*l + 2*q*(5/2)*l + 2*q*3*l = 17ql. The torque from the rods on the left is 2*q*(l/2)*l + 1*q*1*l + 2*q*(3/2)*l + 2*q*2*l + 2*q*(5/2)*l + 1*q*3*l = 17ql. This shows the configuration does not have to be symmetrical to be balanced.
See the linked text file for the step directions for the stable walks for n=6 to n=15.

Scott R. Shannon, Stable walks for n=6 to n=15.

Cf. A116903, A337761, A001411, A001412.

A337761 The number of hanging vertically stable self-avoiding walks of length n on a 3D cubic lattice where both the nodes and connecting rods have mass.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 13, 29, 73, 193, 581, 1717, 7029, 21981, 79625, 248697, 981353, 3275085, 13235333

Offset: 1

Scott R. Shannon, Sep 19 2020

Consider a self-avoiding walk on a 3D cubic lattice where each visited node is given a fixed mass and each node is connected by a rod of another fixed mass. Hang the resulting lattice structure from a string at the first node. This sequence gives the number of walks of length n such that the structure will hang perfectly vertically, and will return to this position if perturbed.
For a walk to be stable requires the torque around the first node to be zero along both the x and y axial directions for both the node and rod masses, and that the overall center of mass of the structure is lower than the first node. As n increases the number of walks satisfying these conditions decreases rapidly. For example the total number of 3D self-avoiding walks on a cubic lattice in the lower four quadrants for n=18 is A116904(18) = 719101961769. The total number of hanging stable walks for n=18 is 3275085, indicating only one in about 220 thousand walks is stable.
The stable walks are the same as in the 2D case, see A335780, up until n=9; the same stable single-plane walks occur but in both the x-z and y-z plane so the total of these walks is twice A335780. From n=10 stable walks can occur which use all three dimensions. For all stable walks it is found that the final node is always directly underneath the starting node. This is not the case if only the node or rod masses are considered.
See A335780 for the equalivalent sequence on a 2D square lattice.

			a(1)-a(5) = 1 as the only stable walk is a walk straight down from the first node.
a(6) = 5. There is one stable walk confined to a single plane:
.
      X-----+
            |
            |
+-----+-----+
|
|
+-----+
.
where 'X' represents the hanging point first node at (0,0,0).
Assuming a mass of p for the nodes, q for the rods, and a length l for the rods, the total torque from the nodes to the right of the first node is 2*p*l, which equals that from the nodes to the left. The total torque for the rods to the right of the first node is 2*q*(1/2)*l + 1*q*1*l = 2ql, which equals that from the rods to the left. The center of mass is at coordinate (0,0,-1). This walk can be taken in 4 ways thus, with the straight down walk, the total number of stable walks is 4+1 = 5.
a(10) = 193. Other than the straight and single plane walks those using three dimensions now occur. An example of such a walk is:
.
        +--------+      z    y
       /        /        \  /
      /   X-------+       \/
     /        /    \       +-----x
    /        /      \
   +        +        +
    \      /        /
     \    /        /
      +--/----+   /
        /        /
       +--------+
.
The total rotational torque around the y-axis from the nodes with x>0 is 3*p*l, which equals that from the nodes with x<0. The total rotational torque around the x-axis from the nodes with y>0 is 2*p*l, which equals that from the nodes with y<0. The total rotational torque around the y-axis from the rods with x>0 is 2*q*(1/2)*l + 2*q*1*l = 3ql, which equals that from the rods with x<0.  The total rotational torque around the x-axis from the rods with y>0 is 2*q*(1/2)*l + 1*q*1*l = 2ql, which equals that from the rods with x<0. The center of mass is at coordinate (0,0,-1).
a(11) = 581. An example of an 11 step stable walk where the final node is above the first node:
.
                   +              +
                  /              /|       z
                 /              / |       |  y
                /              /  |       | /
               +              /   |       |/
        +      |   X---------+    |       +-----x
       /|      |       +----------+
      / |      |      /
     /  |      |     /
    /   |      |    /
   +-----------+   /
        +---------+
.
This particular walk would be counted twice as it is also stable if hung from the final node.
See the linked text file for the step directions for the stable walks for n=6 to n=12.

Scott R. Shannon, Stable walks for n=6 to n=12.

Cf. A335780, A001412, A116904, A001411.

A377161 a(n) is the numerator of the probability that a self-avoiding random walk on the cubic lattice is trapped after n steps.

Original entry on oeis.org

8, 38, 637, 9759, 86221819, 28522360751, 583791967829, 1801511107253, 6467456149881773

Offset: 11

Hugo Pfoertner, Oct 20 2024

			8/1953125, 38/9765625, 637/58593750, 9759/976562500, 86221819/4687500000000, 28522360751/1687500000000000, 583791967829/22500000000000000, ...

Plot of a(n)/A377162(n) vs n, using Plot 2.

A377162 are the corresponding denominators.

Cf. A001412, A077817, A077818 (see there for more information), A077819, A077820.

a(n)/A377162(n) = A077818(n) / (5^(n-1) * 3^A077819(n) * 2^A077820(n)).

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Fortran

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

A010575 Number of n-step self-avoiding walks on 4-d cubic lattice.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Fortran

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A116904 Number of n-step self-avoiding walks on the upper 4 octants of the cubic grid starting at origin.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A337023 Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2h X 2h X 2h where the walk starts at the center of the box.

Views

Author

Keywords

Examples

Crossrefs

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A335780 The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where both the nodes and connecting rods have mass.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A337761 The number of hanging vertically stable self-avoiding walks of length n on a 3D cubic lattice where both the nodes and connecting rods have mass.

Views

Author

Keywords