A001668 -id:A001668 - cp's OEIS frontend

A179260 Decimal expansion of the connective constant of the honeycomb lattice.

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

Offset: 1

Jonathan Vos Post, Jul 06 2010

This is the case n=8 of the ratio Gamma(1/n)*Gamma((n-1)/n)/(Gamma(2/n)*Gamma((n-2)/n)). - Bruno Berselli, Dec 13 2012
An algebraic integer of degree 4: largest root of x^4 - 4x^2 + 2. - Charles R Greathouse IV, Nov 05 2014
This number is also the length ratio of the shortest diagonal (not counting the side) of the octagon and the side. This ratio is A121601 for the longest diagonal. - Wolfdieter Lang, May 11 2017 [corrected Oct 28 2020]
From Wolfdieter Lang, Apr 29 2018: (Start)
This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 3, pp. 69-74. See also the comments in A302711 with the Romanus link and his Exemplum tertium.
This problem is equivalent to R(45, 2*sin(Pi/120)) = 2*sin(3*Pi/8) with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/120) see A302715. (End)

			1.84775906502257351225636637879357657364483325172728497223019546256107001500...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.10, p. 333.
Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.
Neal Madras and Gordon Slade, Self-avoiding walks, Probability and its Applications, Birkhäuser Boston, Inc. Boston, MA, 1993.

Hugo Duminil-Copin and Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), arXiv:1007.0575 [math-ph], 2011.
Hugo Duminil-Copin and Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), Ann. Math. 175 (2012), pp. 1653-1665.
Steven Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 5.10; arXiv:2001.00578 [math.HO], 2020.
Pierre-Louis Giscard, Que sait-on compter sur un graphe. Partie 3 (in French), Images des Mathématiques, CNRS, 2020.
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, On the scaling limit of planar self-avoiding walk, Fractal Geometry and applications: a jubilee of Benoit Mandelbrot, Part 2, 339-364. Proc.
Bernard Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys. Rev. Lett. 49 (1982), 1062-1065.
Jonathan Sondow and Huang Yi, New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2)), arXiv:1005.2712 [math.NT], 2010.
Jonathan Sondow and Huang Yi, New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2)), Amer. Math. Monthly 117 (2010) 912-917.
Index entries for algebraic numbers, degree 4.
Index entries for sequences related to Chebyshev polynomials.

Cf. A002193, A047522, A101464, A127672, A144981, A154739, A302715, A121601.

RealDigits[Sqrt[2+Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Jan 19 2014 *)

sqrt(2+sqrt(2)) \\ Charles R Greathouse IV, Nov 05 2014

sqrt(2+sqrt(2)) = (2/1)(6/7)(10/9)(14/15)(18/17)(22/23)... (see Sondow-Yi 2010).
Equals 1/A154739. - R. J. Mathar, Jul 11 2010
Equals 2*A144981. - Paul Muljadi, Aug 23 2010
log (A001668(n)) ~ n log k where k = sqrt(2+sqrt(2)). - Charles R Greathouse IV, Nov 08 2013
2*cos(Pi/8) = sqrt(2+sqrt(2)). See a remark on the smallest diagonal in the octagon above. - Wolfdieter Lang, May 11 2017
Equals also 2*sin(3*Pi/8). See the comment on van Roomen's third problem above. - Wolfdieter Lang, Apr 29 2018
Equals i^(1/4) + i^(-1/4). - Gary W. Adamson, Jul 06 2022
Equals Product_{k>=0} ((8*k + 2)*(8*k + 6))/((8*k + 1)*(8*k + 7)). - Antonio Graciá Llorente, Feb 24 2024
Equals Product_{k>=1} (1 - (-1)^k/A047522(k)). - Amiram Eldar, Nov 22 2024

A266925 Number of unrooted self-avoiding walks of n steps on honeycomb lattice.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 18, 31, 61, 107, 207, 378, 724, 1333, 2537, 4723, 8938, 16674, 31455, 58805, 110642, 206849, 388444, 726236, 1361766, 2544568, 4765466, 8900805, 16652803, 31085427, 58108482, 108417265, 202517545, 377659515, 704989607, 1314105907, 2451689503, 4568030529

Offset: 1

Luca Petrone, Jan 06 2016

Asymptotically approaches (1/12) * A001668.

A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi(2k/n - 1), k = 1..n-1.

Original entry on oeis.org

71, 71, 40, 77, 45, 51, 42, 56, 49, 51, 48, 54

Offset: 3

Hugo Pfoertner, Dec 27 2018

The cases n = 3, 4, and 6 correspond to the usual self-avoiding random walks on the honeycomb net, the square lattice, and the hexagonal lattice, respectively. The other cases n = 5, 7, ... are a generalization using self-avoiding rooted walks similar to those defined in A306175, A306177, ..., A306182. The walk is trapped if it cannot be continued without either hitting an already visited (lattice) point or crossing or touching any straight line connecting successively visited points on the path up to the current point.
The result 71 for n=4 was established in 1984 by Hemmer & Hemmer.
The sequence data are based on the following results of at least 10^9 simulated random walks for each n <= 12, with an uncertainty of +- 0.004 for the average walk length:
   n  length
71.132
70.760 (+-0.001)
40.375
77.150
45.297
51.150
42.049
56.189
48.523
51.486
47.9   (+-0.2)
53.9   (+-0.2)

S. Hemmer, P. C. Hemmer, An average self-avoiding random walk on the square lattice lasts 71 steps, J. Chem. Phys. 81, 584 (1984)
Hugo Pfoertner, Examples of self-trapping random walks.
Hugo Pfoertner, Probability density for the number of steps before trapping occurs, 2018.
Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk.
Alexander Renner, Self avoiding walks and lattice polymers, Diplomarbeit, Universität Wien, December 1994.

Cf. A001668, A001411, A001334, A077482, A306175, A306177, A306178, A306179, A306180, A306181, A306182.

Cf. A122223, A122224, A122226, A127399, A127400, A127401, A300665, A323141, A323560, A323562, A323699.

A334330 Number of endless self-avoiding walks of length 2*n for the honeycomb lattice.

Original entry on oeis.org

1, 6, 18, 60, 210, 726, 2448, 8448, 28818, 98556, 336618, 1150320, 3928944, 13419204, 45828192, 156512220, 534463698, 1825120584, 6232259412, 21281168202, 72666555570, 248124503652, 847224827676, 2892836367066, 9877456541376, 33725891989626

Offset: 0

Michel Marcus, Apr 23 2020

Nathan Clisby, Endless self-avoiding walks, arXiv:1302.2796 [cond-mat.stat-mech], 2013. See Table 4 p. 23.

A001668 counts all self-avoiding walks on the honeycomb lattice, without the "endless" restriction.

Cf. A334322 (square lattice), A334326 (simple cubic lattice), A334331 (triangular lattice), A334332 (union jack lattice), A334333 (body centered cubic lattice), A334334 (face centered cubic lattice).

A006851 Trails of length n on honeycomb lattice.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 96, 186, 360, 696, 1344, 2562, 4872, 9288, 17664, 33384, 63120, 119280, 225072, 423630, 797400, 1499256, 2817216, 5286480, 9918768, 18592080, 34840848, 65228874, 122105496, 228402168, 427176336, 798373662, 1491985800, 2786515176, 5203816992, 9712725234, 18127267800

Offset: 0

N. J. A. Sloane

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

Alois P. Heinz, Table of n, a(n) for n = 0..43
H. Duminil-Copin and S. Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt(2)), arXiv:1007.0575 [math-ph], 2010-2011.
A. J. Guttmann, Lattice trails II: numerical results, J. Phys. A 18 (1985), 575-588.

Cf. A001668.

a:= proc(n) option remember; local v, b;
      if n<2 then return 1 +2*n fi;
      v:= proc() false end: v(1, 0):= true;
      b:= proc(n, d, x, y) local c;
            if v(x, y) then `if`(n>0 or [x, y, d]=[1, 0, 1], 0, 1)
          elif n=0 then 1
          else v(x, y):= true;
               c:= b(n-1, [$2..6, 1][d], x+[0, -1, -1, 0, 1, 1][d],
                                         y+[1, 1, 0, -1, -1, 0][d])+
                   b(n-1, [6, $1..5][d], x+[1, 1, 0, -1, -1, 0][d],
                                         y+[-1, 0, 1, 1, 0, -1][d]);
               v(x, y):= false; c
            fi
          end;
      6*b(n-2, 2, 1, 1)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 08 2011

a[n_] := a[n] = Module[{v, b}, If[n<2, Return[1+2*n]]; v[, ] = False; v[1, 0] = True; b[n0_, d_, x_, y_] := Module[{c}, Which[v[x, y], If[n0>0 || {x, y, d} == {1, 0, 1}, 0, 1], n0 == 0, 1, True, v[x, y] = True; c = b[n0-1, {2, 3, 4, 5, 6, 1}[[d]], x+{0, -1, -1, 0, 1, 1}[[d]], y+{1, 1, 0, -1, -1, 0}[[d]]] + b[n0-1, {6, 1, 2, 3, 4, 5}[[d]], x+{1, 1, 0, -1, -1, 0}[[d]], y+{-1, 0, 1, 1, 0, -1}[[d]]]; v[x, y] = False; c]]; 6*b[n-2, 2, 1, 1]]; Table[Print[a[n]]; a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)

A192871 Number of n-step prudent self-avoiding walks on honeycomb lattice.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 90, 168, 318, 594, 1092, 2004, 3678, 6720, 12210, 22128, 40074, 72372, 130380, 234432, 421128, 755208, 1352328, 2418246, 4320552, 7709898, 13744764, 24477618, 43560444, 77448330, 137602440, 244277016, 433399824, 768379830, 1361530134

Offset: 0

Alois P. Heinz, Jul 11 2011

A prudent walk never takes a step pointing towards a vertex it has already visited. Prudent walks are self-avoiding but not reversible in general.

			This 8-step prudent self-avoiding walk on honeycomb lattice from (S) to (E) is not reversible:
.           o...o       o...o
.          .     .     .     .
.     o...o       4---3       o
.    .     .     /     \     .
.   o       6---5       2...o
.    .     /     .     /     .
.     o...7      (S)--1       o
.    .     \     .     .     .
.   o      (E)..o       o...o
.    .     .     .     .
.     o...o       o...0

Alois P. Heinz, Table of n, a(n) for n = 0..110
Mireille Bousquet-Mélou, Families of prudent self-avoiding walks, DMTCS proc. AJ, 2008, 167-180.
Mireille Bousquet-Mélou, Families of prudent self-avoiding walks, arXiv:0804.4843 [math.CO], 2008-2009.
Enrica Duchi, On some classes of prudent walks, in: FPSAC'05, Taormina, Italy, 2005.

Cf. A001668, A006851, A192208.

i:= n-> max(n, 0)+1: d:= n-> max(n-1, -1):
b:= proc(n, x, y, z, u, v, w) option remember;
    `if`(n=0, 1, `if`(x>y, b(n, y, x, w, v, u, z),
    `if`(min(y, z)<=0 or x=-1,
        b(n-1, d(y), d(z), u, i(v), i(w), x), 0)+
    `if`(min(w, x)<=0 or y=-1,
        b(n-1, d(w), d(x), y, i(z), i(u), v), 0)))
    end:
a:= n-> `if`(n<2, 1 +2*n, 6*b(n-2, -1, -1, 1, 2, 1, -1)):
seq(a(n), n=0..20);

i[n_] := Max[n, 0] + 1; d[n_] := Max[n - 1, -1];
b[n_, x_, y_, z_, u_, v_, w_] := b[n, x, y, z, u, v, w] = If[n == 0, 1, If[x > y, b[n, y, x, w, v, u, z], If[Min[y, z] <= 0 || x == -1, b[n - 1, d[y], d[z], u, i[v], i[w], x], 0] + If[Min[w, x] <= 0 || y == -1, b[n - 1, d[w], d[x], y, i[z], i[u], v], 0]]];
a[n_] := If[n < 2, 1 + 2 n, 6 b[n - 2, -1, -1, 1, 2, 1, -1]];
a /@ Range[0, 34] (* Jean-François Alcover, Sep 22 2019, after Alois P. Heinz *)

cp's OEIS Frontend

A179260 Decimal expansion of the connective constant of the honeycomb lattice.

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A266925 Number of unrooted self-avoiding walks of n steps on honeycomb lattice.

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A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi*(2*k/n - 1), k = 1..n-1.

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A334330 Number of endless self-avoiding walks of length 2*n for the honeycomb lattice.

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A006851 Trails of length n on honeycomb lattice.

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A192871 Number of n-step prudent self-avoiding walks on honeycomb lattice.

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A336758 Number of n-step self-avoiding walks on the honeycomb lattice with no non-contiguous adjacencies.

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A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi(2k/n - 1), k = 1..n-1.