cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Hugo Pfoertner, Jul 05 2021

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A122223, A127399, A127400, A127401, A258206, A266925.
Cf. A346124-A346132 similar to this sequence with other sets of turning angles.

Formula

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

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

Views

Author

Hugo Pfoertner and Markus Sigg, Jul 30 2021

Keywords

Comments

Closed walks are allowed.

Examples

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

Links

Crossrefs

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

Views

Author

Hugo Pfoertner and Markus Sigg, Jul 31 2021

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Views

Author

Hugo Pfoertner and Markus Sigg, Jul 31 2021

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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, +-3*Pi/7, +-5*Pi/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

Views

Author

Hugo Pfoertner and Markus Sigg, Aug 01 2021

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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, +-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.

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

Views

Author

Hugo Pfoertner and Markus Sigg, Aug 01 2021

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A127399, A127400, A127401, A306181, A316201.
Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

A346128 Numbers m such that no self-avoiding walk that can make turns from the set 0, +-Pi/4, +-Pi/2, +-3*Pi/4, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 9, 11, 12, 13, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38
Offset: 1

Views

Author

Hugo Pfoertner and Markus Sigg, Aug 01 2021

Keywords

Comments

Closed walks (see A316198) are allowed, but except for the closed square-shaped walk of length 4 that fits into the same smallest enclosing circle as the smallest open walk of this length, no other closed walk that fits into a smaller enclosing circle than any open walk of the same length is known.

Examples

			See link for illustrations of terms corresponding to diameters D < 3.83.

Links

Crossrefs

Cf. A127399, A127400, A127401, A306178, A316198.
Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

A346129 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/9, +-Pi/3, +-5*Pi/9, +-7*Pi/9, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41
Offset: 1

Views

Author

Hugo Pfoertner and Markus Sigg, Aug 01 2021

Keywords

Comments

Although closed walks (see A316199) would be allowed, no closed walk that fits into a smaller enclosing circle than any open walk of the same length is known.

Examples

			See link for illustrations of terms corresponding to diameters D < 3.50.

Links

Crossrefs

Cf. A127399, A127400, A127401, A306179, A316199.
Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

A346130 Numbers m such that no self-avoiding walk that can make turns from the set 0, +-Pi/5, +-2*Pi/5, +-3*Pi/5, +-4*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, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 40
Offset: 1

Views

Author

Hugo Pfoertner and Markus Sigg, Aug 01 2021

Keywords

Comments

Closed walks (see A316200) are allowed. The only known closed walk that fits into a smaller enclosing circle than any open walk of the same length occurs for length 8.

Examples

			See link for illustrations of terms corresponding to diameters D < 3.592.

Links

Crossrefs

Cf. A127399, A127400, A127401, A306180, A316200.
Cf. A346123-A346132 similar to this sequence with other sets of turning angles.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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