A078797 -id:A078797 - cp's OEIS frontend

A046661 Number of n-step self-avoiding walks on the square lattice with first step specified.

1, 3, 9, 25, 71, 195, 543, 1479, 4067, 11025, 30073, 81233, 220375, 593611, 1604149, 4311333, 11616669, 31164683, 83779155, 224424291, 602201507, 1611140121, 4316653453, 11536599329, 30870338727, 82428196555, 220329372907

Offset: 1

N. J. A. Sloane

Used as the denominator for the mean square displacement of all different self-avoiding n-step walks in A078797. - Hugo Pfoertner, Dec 09 2002

Number of ways a toy snake with n segments can be bent without flipping the snake upside down. Each segment must be perpendicular or parallel with each adjacent segment. A "slither" is a way of writing down the configuration of a snake; starting from the tail, write down which direction the next segment is pointing (R for right, S for straight, L for left). E.g., a snake with 10 segments may have the valid slither RLRRLLRRL, but not RSRRSSLSL.

Hugo Pfoertner, Table of n, a(n) for n = 1..79
G. T. Barkema and S. Flesia, Two-dimensional oriented self-avoiding walks with parallel contacts, J. Stat. Phys. 85 (1996) no 3/4, 363-381. [a(30) to a(34)]
D. Bennet-Wood, J. L. Cardy, S. Flesia, A. J. Guttmann et al., Oriented self-avoiding walks with orientation-dependent interactions, J. Phys. A: Math. Gen. 28 (1995) no 18, 5143-5163. [up to a(29)]
V. Hart, How to Snakes, Youtube Video (2011).

Cf. A001411, A002900, A078797.

(* b = A001411 *) mo = Tuples[{-1, 1}, 2]; b[0] = 1; b[tg_, p_:{{0, 0}}] := b[tg, p] = Block[{e, mv = Complement[Last[p] + #& /@ mo, p]}, If[tg == 1, Length[mv], Sum[b[tg - 1, Append[p, e]], {e, mv}]]];
a[n_] := b[n]/4;
Table[an = a[n]; Print[an]; an, {n, 1, 16}] (* Jean-François Alcover, Nov 02 2018, after Giovanni Resta in A001411 *)

a(n) = A001411(n)/4 = A002900(n)/2.

A242355 Sum of squared end-to-end distances of all n-step self-avoiding walks on the 4-d cubic lattice.

Original entry on oeis.org

8, 128, 1416, 13568, 119960, 1009440, 8205656, 65068352, 506193144, 3879735776, 29378067080, 220265711040, 1637726387096, 12091336503584, 88727095777896, 647661676223168, 4705654523841704, 34049855885188128, 245482626441965048, 1764039730476165824

Offset: 1

Hugo Pfoertner, Aug 16 2014

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..24 [4th column in Table A6 from Clisby article 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.
N. Clisby, R. Liang and G. Slade Self-avoiding walk enumeration via the lace expansion. Tables in machine-readable format.
Hugo Pfoertner, Results for the 4D Self-Trapping Random Walk.

Cf. A010575 corresponding number of walks, A118313 end-to-end distances for cubic lattice, A078797 end-to-end distances for quadratic lattice, A323856, A323857.

A336448 Sum of square displacements over all n-step self-avoiding walks on a 2D square lattice.

Original entry on oeis.org

0, 4, 32, 164, 704, 2716, 9808, 33788, 112480, 364588, 1157296, 3610884, 11108448, 33765276, 101594000, 302977204, 896627936, 2635423124, 7699729296, 22374323436, 64702914336, 186289216332, 534227118960, 1526445330900, 4347038392480, 12341626847324, 34940293640400, 98660244502668

Offset: 0

Scott R. Shannon, Jul 22 2020

nonn

See A001411 for the corresponding number of n-step self-avoiding walks.

			a(1) = 4 as a single step of length 1 can be taken in four ways on the square lattice the sum of square end-to-end displacements is 4*1 = 4.
a(2) = 32. The two 2-step self-avoiding walks with a first step to the right in the first quadrant with their corresponding square displacements are:
.
    +
    | 2    +---+---+ 4
+---+
.
The first walk can be taken in 8 ways on a square lattice, the latter in 4 ways, thus the total displacement over all 2-step walks is 8*2 + 4*4 = 32.
a(3) = 164. The five 3-step self-avoiding walks with a first step to the right in the first quadrant with their corresponding square displacements are:
.
             +
+---+        |         +---+            +
    | 1      + 5       |    5           | 5   +---+---+---+ 9
+---+        |     +---+        +---+---+
         +---+
.
The first four walks can be taken in 8 ways on a square lattice, the last in 4 ways, thus the total displacement over all 3-step walks is 8*1 + 8*5 + 8*5 + 8*5 + 4*9 = 164.

See A001411 and A078797.

A. J. Guttmann, On the critical behavior of self-avoiding walks, J. Phys. A 20 (1987), 1839-1854.
I. Jensen, Series Expansions for Self-Avoiding Walks

Cf. A001411, A078797, A046661.

a(n) = Sum_{k=0..A001411(n)} ( i_k^2 + j_k^2 ) where (i_k, j_k) are the end points of all different self-avoiding n-step walks.

a(n) = 4*A078797(n).

A306648 Decimal expansion of 3 + 36/(5sqrt(3)Pi) + 48/(5*Pi^2).

Original entry on oeis.org

5, 2, 9, 5, 8, 7, 2, 7, 1, 1, 9, 7, 8, 7, 4, 3, 5, 2, 4, 6, 8, 8, 4, 2, 6, 4, 4, 2, 7, 6, 4, 5, 1, 1, 9, 9, 2, 8, 3, 5, 5, 7, 1, 7, 8, 9, 7, 3, 3, 5, 0, 4, 4, 3, 5, 6, 3, 6, 3, 2, 5, 1, 5, 4, 8, 8, 2, 5, 3, 4, 6, 8, 9, 5, 6, 3, 7, 0, 7, 7, 4, 3, 0, 5, 2, 5, 2, 1, 0, 9, 8, 8, 6, 1, 5, 7, 9, 7, 7, 5

Offset: 1

Scott R. Shannon, Mar 03 2019

This is the mean square end-to-end distance of the 3-step self-avoiding walk with full excluded volume in the 2-dimensional continuum.
Take 4 touching circles of diameter 1 which are joined as a chain and each is free to move around its neighbors' perimeters, but no circle can overlap another. This value is the average of the square of the distance from the middle of the first circle to the middle of the fourth circle, averaged over all possible configurations the chain of 4 non-overlapping circles can take.
To derive the exact value consider the 4-circle chain where each circle has diameter 1. Let t1 be the angle between the first and third circles centered at the second circle, and t2 be the angle between the second and fourth circles centered at the third circle. Using the law of cosines and basic geometry one can show the square of the distance between the centers of the first and fourth circles, r2_4, can be written as r2_4 = 4*sin^2(t1/2) - 4*sin(t1/2)*sin(t1/2+t2) + 1. When 2*Pi/3 <= t1 <= Pi the fourth circle cannot overlap the first, so t2 is free to vary over its full range of Pi/3 to 5*Pi/3. When Pi/3 <= t1 < 2*Pi/3 it can overlap, so the lower bound of t2 is restricted to the angle at which the fourth circle touches the first -- this corresponds to t2 = Pi - t1. To average over all the t1 and t2 angles we must therefore calculate the two t1 ranges separately. The required integral for the mean square distance thus becomes  = ( Integrate(r2_4,{t2,Pi/3,5*Pi/3},{t1,2*Pi/3,Pi}) + Integrate(r2_4,{t2,Pi-t1,5*Pi/3},{t1,Pi/3,2*Pi/3}) ) / ( Integrate(1,{t2,Pi/3,5*Pi/3},{t1,2*Pi/3,Pi}) + Integrate(1,{t2,Pi-t1,5*Pi/3},{t1,Pi/3,2*Pi/3}) ), which includes division by the required normalization integrals. Solving this definite integral gives the exact value for  as 3 + 36/(5*sqrt(3)*Pi) + 48/(5*Pi^2). Note that to find the mean distance, and also the mean square distance for the 5-circle chain, requires integration of sqrt(r2_4) which is a non-elementary integral so only numerical approximations are possible -- these values are approximately 2.25134... and 8.27291... respectively.

			5.295872711978743524688426442764511992835571789733504435636...

Scott R. Shannon, The Two Dimensional Polymer Chain - Statics and Dynamics, PhD Thesis, Monash University, Melbourne, Australia, (1997).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Scott R. Shannon, Java code for a 3-step self-avoiding walk in the continuum. This produces a numerical approximation to the exact value.
Index entries for transcendental numbers

Cf. A078797, A306712.

evalf[120](3+36/(5*Pi*sqrt(3))+48/(5*Pi^2)); # Muniru A Asiru, Mar 05 2019

First@ RealDigits[3 + 36/(5 Sqrt[3] Pi) + 48/(5 Pi^2), 10, 105] (* Michael De Vlieger, Mar 11 2019 *)

3+36/(5*sqrt(3)*Pi)+48/(5*Pi^2) \\ Michel Marcus, Mar 04 2019

Terms a(59) and beyond from Andrew Howroyd, Apr 27 2020

A078798 Sum of Manhattan distances over all self-avoiding n-step walks on square lattice. Numerator of mean Manhattan displacement s(n) = a(n)/A046661(n).

Original entry on oeis.org

1, 6, 23, 80, 263, 834, 2569, 7764, 23095, 67910, 197607, 570560, 1635331, 4661026, 13212739, 37296004, 104836893, 293710714, 820132581, 2283926980, 6343214871, 17578257134, 48604029143, 134141458280, 369519394643

Offset: 1

Hugo Pfoertner, Dec 10 2002

A conjectured asymptotic behavior for the mean Manhattan displacement lim n-> infinity a(n)/(A046661(n)*n^(3/4)) = constant is illustrated in "Asymptotic Behavior of Mean Manhattan Displacement" at first link.

			a(3)=23 because 2 of the A046661(3)=9 walks end at Manhattan distance 1: (0,-1),(0,1) and 7 walks end at Manhattan distance 3: (1,-2),(1,2),2*(2,-1),2*(2,1),(3,0); a(3)=2*1+7*3=23 See also "Distribution of end point distance" at first link.

See under A001411.

Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk

Cf. A001411, A046661, A078797.

c Source code of "FORTRAN program for distance counting" available at first link.

a(n) = Sum_{k=1..A046661(n)} (|i_k| + |j_k|) where (i_k, j_k) are the end points of all different self-avoiding n-step walks.

a(1)=1 inserted by Sean A. Irvine, Jul 16 2025

A078799 Sum of square displacements over all self-avoiding walks on square lattice trapped after n steps.

Original entry on oeis.org

1, 6, 35, 150, 627, 2318, 8400, 28624, 96049, 311002, 994899, 3111570, 9638347, 29398762, 88985840, 266359752, 792360385, 2337329116, 6859721431, 20000471236, 58067533570, 167703151726

Offset: 7

Hugo Pfoertner, Dec 26 2002

The mean squared displacement is given by a(n)/A077482(n) See also "Average Euclidean and Squared End Point Distance" at link

			a(9)=35 because the A077482(9)=11 different self-trapping walks stop at 5*(0,1)->d^2=1, 2*(1,2)->d^2=5, 2*(2,1)->d^2=5, (-1,0)->d^2=1 (3,0)->d^2=9. a(9)=5*1+2*5+2*5+1+9=35 See "Enumeration of all short self-trapping walks" at link

Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk

Cf. A077482, A078797, A078800 (corresponding Manhattan distance sum).

c Program for distance counting available at link.

a(26)-a(28) from Sean A. Irvine, Jul 16 2025

A121791 Series expansion for end-to-end distance of self-avoiding walks on the triangular lattice.

Original entry on oeis.org

1, 12, 97, 654, 3977, 22624, 122821, 644082, 3288739, 16440648, 80783857, 391310240, 1872763387, 8870963422, 41647686501, 194014270964, 897639074623, 4127904278590, 18879838654237, 85930246593928

Offset: 1

N. J. A. Sloane, Aug 30 2006

nonn

I. Jensen, Table of n, a(n) for n = 1..40 [from the Jensen link below]
I. Jensen, Series Expansions for Self-Avoiding Walks

Cf. A001334 corresponding number of self-avoiding walks, A078797 series expansion for end-to-end distances of self-avoiding walks on the square lattice.

Name corrected by Hugo Pfoertner, Aug 11 2014

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A046661 Number of n-step self-avoiding walks on the square lattice with first step specified.

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A242355 Sum of squared end-to-end distances of all n-step self-avoiding walks on the 4-d cubic lattice.

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A336448 Sum of square displacements over all n-step self-avoiding walks on a 2D square lattice.

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A306648 Decimal expansion of 3 + 36/(5*sqrt(3)*Pi) + 48/(5*Pi^2).

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A078798 Sum of Manhattan distances over all self-avoiding n-step walks on square lattice. Numerator of mean Manhattan displacement s(n) = a(n)/A046661(n).

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A078799 Sum of square displacements over all self-avoiding walks on square lattice trapped after n steps.

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A121791 Series expansion for end-to-end distance of self-avoiding walks on the triangular lattice.

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A306648 Decimal expansion of 3 + 36/(5sqrt(3)Pi) + 48/(5*Pi^2).