A002931 -id:A002931 - cp's OEIS frontend

A056639 Number of polygons of length 2n with 3 (self-avoiding polygon) holes on square lattice (not allowing rotations).

62, 3920, 130560, 3058544, 56925214, 900355884, 12619939046, 161208191596, 1913821820768, 21416926011756, 228320059956022, 2337581305737496, 23128907576991302, 222266054948571020, 2082868947950189248

Offset: 16

N. J. A. Sloane, Aug 30 2000

nonn

I. Jensen, Table of n, a(n) for n = 16..45 (from link below)
I. Jensen, More terms

Cf. A002931.

A088702 Number of polygons with polygonal holes on the square lattice enumerated by half-perimeter.

Original entry on oeis.org

0, 1, 2, 7, 28, 124, 588, 2939, 15292, 82168, 453376, 2558074, 14712038, 86029132, 510455002, 3068304865, 18658787150, 114663168405, 711391109162, 4452321247688, 28090360338572, 178550339417087, 1142799275636690

Offset: 1

Markus Voege (markus.voege(AT)inria.fr), Nov 23 2003

nonn

The polygons and the hole are self-avoiding and mutually-avoiding, i.e., no degree four vertices are allowed. Translations are allowed, rotations and reflections are not allowed. The contribution of the holes to the perimeter is counted. The number of the holes is not limited, possibly no holes.

A. J. Guttmann, I. Jensen, L. H. Wong and I. G. Enting, J. Phys. A, Vol. 33 (2000) 1735-1764.

I. Jensen, Table of n, a(n) for n = 1..43 (from link below)
I. Jensen, More terms
I. Jensen, Series expansions for self-avoiding polygons

Cf. A002931 (self-avoiding polygons), A056634 (self-avoiding polygons with exactly one hole), A056638 (self-avoiding polygons with exactly two holes), A056639 (self-avoiding polygons with exactly three holes).

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

A056621 a(n) is the total radius of gyration of all self-avoiding polygons of length 2n on the square lattice.

Original entry on oeis.org

8, 66, 600, 5164, 42872, 346828, 2754056, 21549780, 166626744, 1275865332, 9690096824, 73090383120, 548064459968, 4088719617824, 30367415294800, 224659143155964, 1656259765448200, 12172580326973688

Offset: 2

N. J. A. Sloane, Aug 30 2000

nonn

I. Jensen, Table of n, a(n) for n = 2..50 (from link below)
Sean A. Irvine, Java program (github)
I. Jensen, More terms

Cf. A002931.

Title clarified by Sean A. Irvine, Apr 25 2022

A056633 a(n) is the total squared perimeter of all self-avoiding polygons of area n on the square lattice.

Original entry on oeis.org

16, 72, 384, 1864, 8720, 40064, 180992, 805416, 3542720, 15437280, 66739168, 286614344, 1223894208, 5200593136, 22003683264, 92745730384, 389613088944, 1631810452416, 6816075114112, 28401442383848

Offset: 1

N. J. A. Sloane, Aug 30 2000

nonn

I. Jensen, Table of n, a(n) for n = 1..40 (from link below)
I. Jensen, More terms

Cf. A002931, A056632.

Title clarified by Sean A. Irvine, Apr 26 2022

A060379 Number of self-avoiding polygons on the 2-dimensional square lattice with perimeter 2n with at most 4 horizontal edges in each vertical cross-section.

Original entry on oeis.org

1, 2, 7, 28, 124, 588, 2938, 15266, 81770, 448698, 2510813, 14277838, 82286365, 479610362, 2822332127, 16745262798

Offset: 2

Doron Zeilberger, Apr 03 2001

			a(3) = 2 because there are 2 self-avoiding polygons of perimeter 2*3 with at most 4 horizontal edges per vertical cross-section.

Doron Zeilberger, The Umbral Transfer-Matrix Method. IV. Counting Self-AvoidingPolygons and walks; Local copy [Pdf file only, no active links]

Cf. A005435, A002931.

See Appendix 2 of the reference (a 7-page system of linear functional equations for 5 unknown generating functions, one of which is the desired generating function).

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.

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

A003305 Figure 8's with 2n edges on the square lattice.

Original entry on oeis.org

2, 8, 40, 208, 1120, 6200, 35236, 204664, 1210392, 7266824, 44184288

Offset: 4

N. J. A. Sloane

Rotations and reflections are distinct. 2n is the total number of edges in the figure. Each "figure 8" is two of the polygons counted by A002931 intersecting at a single point. Like A003304 but now the smaller loop need not be a simple square. - Sean A. Irvine, Mar 21 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. G. Whittington and J. P. Valleau, Figure eights on the square lattice: enumeration and Monte Carlo estimation, J. Phys. A 3 (1970), 21-27.

Cf. A002931, A003304.

a(11)-a(14) from Sean A. Irvine, Mar 21 2017

A156816 Decimal expansion of the positive root of the equation 13x^4 - 7x^2 - 581 = 0.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Feb 16 2009

This constant approximates the connective constant of the square lattice, which is known only numerically, but "no derivation or explanation of this quartic polynomial is known, and later evidence has raised doubts about its validity" [Bauerschmidt et al, 2012, p. 4]. - Andrey Zabolotskiy, Dec 26 2018

			x = 2.63815853034174086843...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.10, p. 331.
N. Madras and G. Slade, The Self-Avoiding Walk (Boston, Birkhauser), 1993.

Roland Bauerschmidt, Hugo Duminil-Copin, Jesse Goodman, and Gordon Slade, Lectures on Self-Avoiding Walks, arXiv:1206.2092 [math.PR], 2012.
M. Bousquet-Mélou, A. J. Guttmann and I. Jensen, Self-avoiding walks crossing a square, arXiv:cond-mat/0506341, 2005.
Pierre-Louis Giscard, Que sait-on compter sur un graphe. Partie 3 (in French), Images des Mathématiques, CNRS, 2020.
Jesper Lykke Jacobsen, Christian R. Scullard, and Anthony J. Guttmann, On the growth constant for square-lattice self-avoiding walks, J. Phys. A: Math. Theor., 49 (2016), 494004; arXiv:1607.02984 [cond-mat.stat-mech], 2016.
Index entries for algebraic numbers, degree 4.

Cf. A001411, A002931, A179260, A249776.

RealDigits[Sqrt[1/26*(7+Sqrt[30261])],10,120][[1]] (* Harvey P. Dale, Nov 22 2014 *)

polrootsreal(13*x^4-7*x^2-581)[2] \\ Charles R Greathouse IV, Apr 16 2014

x = sqrt(7/26 + sqrt(30261)/26).

cp's OEIS Frontend

A056639 Number of polygons of length 2n with 3 (self-avoiding polygon) holes on square lattice (not allowing rotations).

Views

Author

Keywords

Links

Crossrefs

A088702 Number of polygons with polygonal holes on the square lattice enumerated by half-perimeter.

Views

Author

Keywords

Comments

References

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A056621 a(n) is the total radius of gyration of all self-avoiding polygons of length 2n on the square lattice.

Views

Author

Keywords

Links

Crossrefs

Extensions

A056633 a(n) is the total squared perimeter of all self-avoiding polygons of area n on the square lattice.

Views

Author

Keywords

Links

Crossrefs

Extensions

A060379 Number of self-avoiding polygons on the 2-dimensional square lattice with perimeter 2n with at most 4 horizontal edges in each vertical cross-section.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Formula

Extensions

A003305 Figure 8's with 2n edges on the square lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A156816 Decimal expansion of the positive root of the equation 13x^4 - 7x^2 - 581 = 0.

Views

Author