A127400 -id:A127400 - cp's OEIS frontend

A127399 Number of segments of the longest possible zigzag paths fitting into a circle of diameter 2 if the path with index n is constructed according to the rules of the "Snakes on a Plane" problem of Al Zimmermann's programming contest.

2, 6, 4, 6, 7, 7, 8, 11, 9, 11, 12, 14, 13, 17, 16, 19, 20, 20, 23, 23, 23, 27, 27, 28, 29

Offset: 2

Hugo Pfoertner, Jan 12 2007

The extension of the contest problem to larger sets of hinge angles was proposed by James R. Buddenhagen. A link to the contest rules is given in A127400. Results up to n=32 were found by Markus Sigg. Known lower bounds for the next terms are a(27)>=29, a(28)>=32, a(29)>=34, a(30)>=34, a(31)>=34, a(32)>=39.

Hugo Pfoertner, Visualization of longest zigzag paths fitting into circle of diameter 2.

Cf. A127400 [solutions for container diameter 3], A127401 [solutions for container diameter 4], A122223, A122224, A122226 [solutions for hinge angles excluded from contest].

A127401 Number of segments of the longest possible zigzag paths fitting into a circle of diameter 4 if the path with index n is constructed according to the rules of the "Snakes on a Plane" problem of Al Zimmermann's programming contest.

Original entry on oeis.org

12, 14, 33, 19, 43, 44, 59, 50

Offset: 3

Hugo Pfoertner, Jan 12 2007

Links to the contest rules and to visualizations of the results are given in A127400. Known lower bounds for the next terms are a(10)>=50, a(11)>=77, a(12)>=71. [updated by Hugo Pfoertner, May 21 2011]

Hugo Pfoertner, Data supporting a(10)=50, October 2020.

Cf. A127399 [solutions for container diameter 2], A127400 [solutions for container diameter 3], A122223, A122224, A122226 [solutions for hinge angles excluded from contest].

a(10) from Hugo Pfoertner, Nov 01 2020

A346123 Numbers m such that no self-avoiding walk of length m + 1 on the honeycomb net fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 2, 6, 7, 10, 12, 13, 14, 15, 16, 23, 24, 25, 27, 28, 30, 33, 36, 37, 38, 42, 43, 46, 53, 54, 55, 56, 58, 59, 62

Offset: 1

Hugo Pfoertner, Jul 05 2021

The segments of the walk can make relative turns of +- 60 degrees. The walks may be open or closed.

			Illustration of initial terms:
                               %%% %%% %%%
                           %                %
                         %                    %
      %  %              %                     /%
   %        %          %      a(2) = 2       /  %
  %__________%        %                     /    %
  %   L = 1  %       %                     /      %
   %  D = 1 %        %   L = 2, D = 1.732 /       %
      %  %           %                   /        %
                      %                 / Pi/3   %
    a(1) = 1           %-------------- .  .  . .%
                        %                      %
                          %                  %
                              %%% %%% %%%
.
           %%% %%%% %%%                         %%% %%%% %%%
        %                %                   %                %
      %                    %               %                  \ %
     %                      %             %                    \ %
    %                        %           %                      \ %
   %                          %         %                        \ %
  %                            %       %                          \ %
  %.      L = 3, D = 2.00     .%       %.      L = 4, D = 2.00     .%
  % \                        / %       % \                        / %
   % \                      / %         % \                      / %
    % \                    / %           % \                    / %
     % \                  / %             % \                  / %
       % ---------------- %                 % ---------------- %
           %%% %%% %%%                          %%% %%% %%%
.
            %%% %%% %%%                          %%% %%% %%%
        % ______________ %                   % ______________ %
      %                  \ %               % /                \ %
     %                    \ %             % /                  \ %
    %                      \ %           % /                    \ %
   %                        \ %         % /       a(3) = 6       \ %
  %                          \ %       % /                        \ %
  %.      L = 5, D = 2.00     .%       %.      L = 6, D = 2.00     .%
  % \                        / %       % \                        / %
   % \                      / %         % \                      / %
    % \                    / %           % \                    / %
     % \                  / %             % \                  / %
       % ---------------- %                 % ---------------- %
           %%% %%%% %%%                         %%% %%%% %%%
.
The path of minimum diameter of length 7 requires an enclosing circle of D = 3.055, which is greater than the previous minimum diameter of D = 2.00 corresponding to a(3) = 6. No path of length 8 exists that fits into a circle of D = 3.055, thus a(4) = 7.
See link for illustrations of terms corresponding to diameters D <= 9.85.

Hugo Pfoertner, Examples of paths of maximum length.

Cf. A122223, A127399, A127400, A127401, A258206, A266925.

Cf. A346124-A346132 similar to this sequence with other sets of turning angles.

a(n+1) >= a(n) + 1 for n > 1; a(1) = 1.

A346132 Numbers m such that no self-avoiding walk that can make turns from the set 0, +-Pi/6, +-2Pi/6, +-3Pi/6, +-4Pi/6, +-5Pi/6, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 29, 30, 32, 33, 34, 35, 36, 37, 41

Offset: 1

Hugo Pfoertner and Markus Sigg, Jul 21 2021

Closed walks are allowed (see A316192). The only known closed walks that fit into a smaller enclosing circle than any open walk of the same length occur for lengths 3 and 5.

			See link for illustrations of terms corresponding to diameters D < 3.23.

Hugo Pfoertner, Examples of paths of maximum length.

Cf. A127399, A127400, A127401, A306182, A316192.

Cf. A346123-A346131 similar to this sequence with other sets of turning angles.

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.

A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 15, 16, 18, 20, 21, 23, 24, 25, 26, 27, 28, 32, 34, 36, 38, 44, 46, 48, 52, 56, 58, 60

Offset: 1

Hugo Pfoertner and Markus Sigg, Jul 30 2021

Closed walks are allowed.

			See link for illustrations of terms corresponding to diameters D < 8.5.

Hugo Pfoertner, Examples of paths of maximum length.

The squared radii of the enclosing circles are a subset of A192493/A192494.
Cf. A037245, A122224, A127399, A127400, A127401, A266549, A316194, A346993, A346994, A346995.
Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

A346126 Numbers m such that no self-avoiding walk of length m + 1 on the hexagonal lattice fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 10, 12, 14, 15, 16, 19, 20, 22, 23, 24, 25, 27, 31, 32, 34, 37, 38, 39, 40, 42, 43, 44, 45, 48, 49, 55, 56, 57, 58, 60, 61

Offset: 1

Hugo Pfoertner and Markus Sigg, Jul 31 2021

Open and closed walks are allowed. It is conjectured that all optimal paths are closed except for the trivial path of length 1. See the related conjecture in A122226.

			See link for illustrations of terms corresponding to diameters D <= 8.

Hugo Pfoertner, Examples of paths of maximum length.

Cf. A122226, A125852, A127399, A127400, A127401, A151541, A284869, A306176, A316196.
Cf. A346123 (similar to this sequence, but for honeycomb net), A346124 (ditto for square lattice).
Cf. A346125, A346127-A346132 (similar to this sequence, but with other sets of turning angles).

A346125 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/5, +-3*Pi/5, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 14, 15, 17, 19, 25, 27, 28, 31, 32, 33, 34, 35, 37, 38, 39, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 57, 59

Offset: 1

Hugo Pfoertner and Markus Sigg, Jul 31 2021

Although closed walks (see A316195) would be allowed, no closed walk that fits into a smaller enclosing circle than any open walk of the same length is known.

			See link for illustrations of terms corresponding to diameters D < 5.114.

Hugo Pfoertner, Examples of paths of maximum length.

Cf. A127399, A127400, A127401, A306175, A316195.

Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

A346127 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/7, +-3Pi/7, +-5Pi/7, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 31, 35, 37, 38, 39, 40, 41, 42, 43, 45, 47, 48

Offset: 1

Hugo Pfoertner and Markus Sigg, Aug 01 2021

Although closed walks (see A316197) would be allowed, no closed walk that fits into a smaller enclosing circle than any open walk of the same length is known.

			See link for illustrations of terms corresponding to diameters D < 4.126.

Hugo Pfoertner, Examples of paths of maximum length.

Cf. A127399, A127400, A127401, A306177, A316197.

Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

A346131 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/11, +-3Pi/11, +-5Pi/11, +-7Pi/11, +-9Pi/11, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41

Offset: 1

Hugo Pfoertner and Markus Sigg, Aug 01 2021

Although closed walks (see A316201) would be allowed, no closed walk that fits into a smaller enclosing circle than any open walk of the same length is known.

			See link for illustrations of terms corresponding to diameters D < 3.53.

Hugo Pfoertner, Examples of paths of maximum length.

Cf. A127399, A127400, A127401, A306181, A316201.

Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

cp's OEIS Frontend

A127399 Number of segments of the longest possible zigzag paths fitting into a circle of diameter 2 if the path with index n is constructed according to the rules of the "Snakes on a Plane" problem of Al Zimmermann's programming contest.

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A127401 Number of segments of the longest possible zigzag paths fitting into a circle of diameter 4 if the path with index n is constructed according to the rules of the "Snakes on a Plane" problem of Al Zimmermann's programming contest.

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A346123 Numbers m such that no self-avoiding walk of length m + 1 on the honeycomb net fits into the smallest circle that can enclose a walk of length m.

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A346132 Numbers m such that no self-avoiding walk that can make turns from the set 0, +-Pi/6, +-2*Pi/6, +-3*Pi/6, +-4*Pi/6, +-5*Pi/6, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

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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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A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

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A346126 Numbers m such that no self-avoiding walk of length m + 1 on the hexagonal lattice fits into the smallest circle that can enclose a walk of length m.

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A346125 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/5, +-3*Pi/5, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

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A346127 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/7, +-3*Pi/7, +-5*Pi/7, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

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A346131 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/11, +-3*Pi/11, +-5*Pi/11, +-7*Pi/11, +-9*Pi/11, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

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A346132 Numbers m such that no self-avoiding walk that can make turns from the set 0, +-Pi/6, +-2Pi/6, +-3Pi/6, +-4Pi/6, +-5Pi/6, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

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.

A346127 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/7, +-3Pi/7, +-5Pi/7, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

A346131 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/11, +-3Pi/11, +-5Pi/11, +-7Pi/11, +-9Pi/11, of length m + 1 fits into the smallest circle that can enclose a walk of length m.