A010566 -id:A010566 - cp's OEIS frontend

A336265 Number of 2D closed-loop self-avoiding paths on a square lattice where each path consists of steps with successive lengths equal to the prime numbers, from 2 to prime(2n+1).

0, 0, 0, 0, 0, 0, 56, 64, 448, 1552, 8952, 65120, 284584, 1491800, 8467816, 48961856, 307751136, 1781258728

Offset: 0

Scott R. Shannon, Jul 15 2020

This sequence gives the number of closed-loop self avoiding walks on a 2D square lattice where the walk consists of steps with successive lengths equal to the prime numbers. No closed loop path is possible until n = 6, i.e. prime(13) = 41. This walk consists of steps of length 2,3,5,7,11,13,17,19,23,29,31,37,41.

Similar to A010566, where only an even number of steps can form a closed loop, here only an odd number can. This is due to the requirement that the total distance stepped in each of the four directions away from the origin must be matched by an equal distance in the opposite direction. As all primes, other than 2, are odd and unique, this can only occur if the total number of steps in a given direction (other than the direction of the first step of length 2) is even. However the first single step of length 2 still requires an even number of odd length steps to return to the origin, giving an odd number of steps overall in that direction. Adding up the four directions gives an overall odd number for the total number of steps.

			a(0) to a(5) = 0 as no closed-loop walk is possible.
a(6) = 56. There are seven walks which form closed loops when considering only those which start with one or more steps to the right followed by a step upward. These walks consist of steps with lengths 2,3,5,7,11,13,17,19,23,29,31,37,41. See the attached linked text file for the images. Each of these can be walked in eight ways on a 2D square lattice, giving a total number of closed loops of 7*8 = 56.
See the attached linked text files for images of n = 7 and n = 8.

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.
Scott R. Shannon, Images for closed-loops for n = 6, maximum prime = 41.
Scott R. Shannon, Images for closed-loops for n = 7, maximum prime = 47.
Scott R. Shannon, Images for closed-loops for n = 8, maximum prime = 59.

Cf. A010566, A334720, A335305, A334877, A334602.

A019266 Cycle class sequence c(2n) (the number of true cycles of length 2n in which a certain node is included) for square lattice.

Original entry on oeis.org

1, 0, 4, 12, 56, 280, 1488, 8232, 47008, 274824, 1636520, 9890584, 60510480, 374019776, 2332131872, 14651535840, 92653845120, 589317728376, 3767523372432, 24196006128592, 156030800105840, 1009911004804296, 6558631830442384, 42723991459518368, 279091277437885920

Offset: 0

Georg Thimm (mgeorg(AT)ntu.edu.sg)

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..65
G. Thimm, Cycle sequences of crystal structures
G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.

Cf. A002931, A010566.

(Length@FindCycle[{NearestNeighborGraph[Tuples[Range[2 # + 4], 2], {All, 1.}], {#+2,#+2}}, {2 #}, All]) &  /@ Range[11] (* Gabriel B. Apolinario, Jan 07 2017 *)

a(n) = A010566(n) / 2 = 2 * n * A002931(n) for n > 0. - Andrey Zabolotskiy, Jul 26 2022

a(11) from Gabriel B. Apolinario, Jan 07 2017
a(12)-a(18) from Aleksandr D. Krotov, Jan 07 2018
a(19)-a(20) from Aleksandr D. Krotov, Mar 22 2019
Terms a(21) and beyond added by Andrey Zabolotskiy, Jul 26 2022, using A002931

A225877 Number of (2n-1)-step self-avoiding paths between two adjacent sites of a 2-dimensional square lattice.

Original entry on oeis.org

1, 2, 6, 28, 140, 744, 4116, 23504, 137412, 818260, 4945292, 30255240, 187009888, 1166065936, 7325767920, 46326922560, 294658864188, 1883761686216, 12098003064296, 78015400052920, 504955502402148, 3279315915221192, 21361995729759184, 139545638718942960

Offset: 1

Felix A. Pahl, May 19 2013

nonn

For n > 1, a(n) = A010566(n)/4: every self-avoiding open path from P to an adjacent site Q (except the one for n = 1) can be completed to a self-avoiding closed path by adding an edge from Q back to P, and exactly 1/4 of all closed paths through P contain that edge.

Felix A. Pahl, Table of n, a(n) for n = 1..55

A002931 = Cases[Import["https://oeis.org/A002931/b002931.txt", "Table"], {, }][[All, 2]]; a[n_] := n A002931[[n]];
a /@ Range[55] (* Jean-François Alcover, Jan 11 2020 *)

For n>1, a(n) = n*A002931(n) = A010566(n)/4.

A342800 Number of self-avoiding polygons on a 3-dimensional cubic lattice where each walk consists of steps with incrementing length from 1 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 24, 72, 0, 0, 1704, 5184, 0, 0, 193344, 600504, 0, 0, 34321512, 141520752, 0, 0, 9205815672, 37962945288, 0, 0

Offset: 1

Scott R. Shannon, Mar 21 2021

This sequence gives the number of self-avoiding polygons (closed-loop self-avoiding walks) on a 3D cubic lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. Like A334720 and A335305 only n values corresponding to even triangular numbers can form closed loops. All possible paths are counted, including those that are equivalent via rotation and reflection.

			a(1) to a(6) = 0 as no self-avoiding closed-loop walk is possible.
a(7) = 24 as there is one walk which forms a closed loop which can be walked in 24 different ways on a 3D cubic lattice. These walks, and those for n(8) = 72, are purely 2-dimensional. See A334720 for images of these walks.
a(11) = 1704. These walks consist of 120 purely 2-dimensional walks and 1584 3-dimensional walks. One of these 3-dimensional walks is:
.
                                /|
                               / |                        z  y
                              /  |                        | /
                        7 +y /   |                        |/
                            /    | 8 -z                   |----- x
             6 +x          /     |
  |---.---.---.---.---.---/      |               9 +x
  |                              |---.---.---.---.---.---.---.---.---/
  | 5 +z                                                            /
  |                                                                /
  |---.---.---.---/                                               /
        4 -x     /  3 +y                                         /
                /                                               /  10 -y
                | 2 +z                                         /
                |                                             /
                | 1 +z                                       /
                X---.---.---.---.---.---.---.---.---.---.---/
                                     11 -x
.

A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

Cf. A334720, A334877, A342807, A001413, A002896, A002899, A010566, A335305.

A345676 Number of closed-loop self-avoiding paths on a 2-dimensional square lattice where each path consists of steps with successive lengths equal to the square numbers, from 1 to n^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 368, 264, 0, 0, 1656, 5104, 0, 0, 62016, 105344, 0, 0, 1046656, 3181104

Offset: 1

Scott R. Shannon, Sep 04 2021

This sequence gives the number of closed-loop self-avoiding walks on a 2D square lattice where the walk starts with a step length of 1 which then increments at each step to the next square number until the step length is n^2. No closed-loop path is possible until n = 15.
Like A334720 and A335305 the only n values that can form closed loop walks are those which correspond to the indices of even triangular numbers. Curiously though n = 16 walks form no closed loops, even though both n = 15 and n = 16 are indices of such numbers.
As in A010566 all possible paths are counted, including those that are equivalent via rotation and reflection.

			a(1) to a(14) = 0 as no closed-loop paths are possible.
a(15) = 32 as there are four different paths which form closed loops, and each of these can be walked in eight different ways on a 2D square lattice. These walks consist of steps with lengths 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. See the linked text images.

A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.
Scott R. Shannon, Images of the closed loops for n = 15. The line lengths in this text file are long so it may need to be downloaded to be viewed correctly.

Cf. A347506, A334720, A336265, A337550, A010566, A000217.

A364781 Triangular array read by rows: T(n, k) is the number of zero-energy states from the partition function in the Ising model for a finite n*k square lattice with periodic boundary conditions.

Original entry on oeis.org

0, 2, 12, 0, 26, 0, 2, 100, 1346, 20524, 0, 322, 0, 272682, 0, 2, 1188, 72824, 3961300, 226137622, 13172279424, 0, 4258, 0, 58674450, 0, 777714553240, 0, 2, 15876, 3968690, 876428620, 199376325322, 46463664513012, 10990445640557042, 2627978003957146636, 0, 59138, 0, 13184352554, 0, 2799323243348702, 0, 633566123999182005386, 0

Offset: 1

Thomas Scheuerle, Aug 07 2023

Imagine an n X k square tiling on a 2D surface with torus topology. T(n, k) is the number of ways two colors can be assigned to all tiles such that the overall length of the boundary between the colored regions is n*k.

The number of solutions with the additional constrain that exactly k tiles must have the lesser represented color is given for tilings with size 2 X 2*k by A241023(k). In the case 2 X 2*k is k also the minimum count of tiles with the same color in all solutions.

			Triangle begins:
  0;
  2,    12;
  0,    26,       0;
  2,   100,    1346,     20524;
  0,   322,       0,    272682,            0;
  2,  1188,   72824,   3961300,    226137622,    13172279424;
  0,  4258,       0,  58674450,            0,   777714553240,                 0;
  2, 15876, 3968690, 876428620, 199376325322, 46463664513012, 10990445640557042, 2627978003957146636;
  ...

Manuel Kauers, Triangular array flattened. Table of n, a(n) for n = 1..120
Roland Häggkvist, Anders Rosengren, Daniel Andrén, Petras Kundrotas, Per Håkan Lundow, and Klas Markström, Computation of the Ising partition function for 2-dimensional square grids, Phys. Rev. E 69, 046104 (April 16 2004).
Manuel Kauers, Onsager's solution of the Ising model could have been guessed, presentation slides (2018).
Thomas Scheuerle, Some values for T(k, k) from Klas Markström and R. Häggkvist et al., extracted from calculation results provided with their work. (See link.)

Cf. A001411, A002931, A010566, A241023.

function a = A364781( n, k )
    a = 0;
    for m = 1:2^(n*k)-2
        if isingSum( reshape(1-2*bitget(m,1:n*k),n ,k)) == 0
            a = a + 1;
        end
    end
end
function e = isingSum( config )
    e = 0; si = size(config);
    for j = 1:si(2)
        for k = 1:si(1)
            S = config(k, j);
            nb = config(1+mod(k , si(1)), j) + config(k, 1+mod(j , si(2)));
            e = e + (-nb)*S;
        end
    end
end

T(n, k) = 0 if n*k is odd.

a(27) - a(45) from Manuel Kauers, Sep 07 2023

A348334 Table read by downward antidiagonals: T(n,k) is the number of self-avoiding walks on a 2D square lattice for a chain growing to total length n after taking k steps (see Comments lines).

Original entry on oeis.org

4, 12, 4, 36, 12, 4, 108, 36, 12, 4, 324, 108, 36, 12, 4, 972, 324, 108, 36, 12, 4, 2916, 972, 324, 100, 36, 12, 4, 8748, 2916, 972, 284, 100, 36, 12, 4, 26244, 8748, 2916, 804, 284, 100, 36, 12, 4, 78732, 26244, 8748, 2276, 804, 284, 100, 36, 12, 4

Offset: 1

Scott R. Shannon, Oct 13 2021

Consider a chain starting at the origin of a 2D square lattice with an initial length of one and where after each step it grows by one unit in length up to a maximum length of n. Like a standard self-avoiding walk it cannot visit any lattice coordinate it already occupies. After k steps, where k > n, the tail of the chain moves away from the origin as the head of the chain continues to move to all unoccupied coordinates. This means that the chain can eventually revisit the origin when it has taken more than n steps as the tail of the chain no longer occupies that coordinate. In general if a coordinate is visited after m steps then it can be revisited on step m + n + 1 or beyond. This sequence lists the total number of possible walks for a chain growing to maximum length n, with n>=1, after it has taken k steps, with k>= 1.

			For n = 1, 2, 3 the total number of walks is the same as the non-backtracking random walk of A003946 as the chain can never intersect itself.
For n = 4 and beyond for small k the number of walks is the same as the standard 2D SAW of A001411 as for k<=n the chain has not moved away from the origin or any previously visited coordinate. However for k>n and beyond previously visited coordinates become free to move to so the number of possible walks is more than A001411. The first time this happens is for a(4,6):
.
        *---*---*
        |       ^
        *---X > +
.
The arrows indicate the direction of the walk on its first and second step. By time the sixth step occurs the origin, marked with an 'X', and the coordinate at (1,0), are unoccupied, thus the chain is able to step back to the origin, something not possible in A001411. If the walk starts with one or more steps to the right followed by an upward step this can occur in three ways. These walks can be performed in eight total ways on a 2D lattice so that total number of such walks is 8*3 = 24. Therefore a(4,6) = A001411(6) + 24 = 780 + 24 = 804.
.
The table begins:
.
4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, ...
4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, ...
4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, ...
4, 12, 36, 100, 284, 804, 2276, 6444, 18244, 51652, 146236, 414020, 1172164, ...
4, 12, 36, 100, 284, 804, 2276, 6444, 18244, 51652, 146236, 414020, 1172164, ...
4, 12, 36, 100, 284, 780, 2172, 6028, 16732, 46436, 128892, 357748,  992964, ...
4, 12, 36, 100, 284, 780, 2172, 6028, 16732, 46436, 128892, 357748,  992964, ...
4, 12, 36, 100, 284, 780, 2172, 5916, 16268, 44660, 122596, 336428,  923316, ...
4, 12, 36, 100, 284, 780, 2172, 5916, 16268, 44660, 122596, 336428,  923316, ...
4, 12, 36, 100, 284, 780, 2172, 5916, 16268, 44100, 120292, 327908,  893788, ...
4, 12, 36, 100, 284, 780, 2172, 5916, 16268, 44100, 120292, 327908,  893788, ...
4, 12, 36, 100, 284, 780, 2172, 5916, 16268, 44100, 120292, 324932,  881500, ...
4, 12, 36, 100, 284, 780, 2172, 5916, 16268, 44100, 120292, 324932,  881500, ...
...

Cf. A003946, A001411, A010566.

row(1..3,k) = A003946(k);

row(n,k) = A001411(k) for k <= n.

cp's OEIS Frontend

A336265 Number of 2D closed-loop self-avoiding paths on a square lattice where each path consists of steps with successive lengths equal to the prime numbers, from 2 to prime(2n+1).

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A019266 Cycle class sequence c(2n) (the number of true cycles of length 2n in which a certain node is included) for square lattice.

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A225877 Number of (2n-1)-step self-avoiding paths between two adjacent sites of a 2-dimensional square lattice.

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Mathematica

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A342800 Number of self-avoiding polygons on a 3-dimensional cubic lattice where each walk consists of steps with incrementing length from 1 to n.

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A345676 Number of closed-loop self-avoiding paths on a 2-dimensional square lattice where each path consists of steps with successive lengths equal to the square numbers, from 1 to n^2.

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A364781 Triangular array read by rows: T(n, k) is the number of zero-energy states from the partition function in the Ising model for a finite n*k square lattice with periodic boundary conditions.

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A348334 Table read by downward antidiagonals: T(n,k) is the number of self-avoiding walks on a 2D square lattice for a chain growing to total length n after taking k steps (see Comments lines).

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