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 1-5 of 5 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.

A346993 Record numbers of grid points in a square lattice covered by a continuously growing circular disk if the center of the disk is chosen to cover the maximum possible number of grid points.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 9, 12, 13, 14, 16, 17, 21, 22, 24, 26, 27, 28, 32, 33, 37, 38, 39, 40, 41, 44, 45, 46, 47, 48, 52, 56, 57, 58, 59, 61, 62, 63, 64, 65, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 89, 90, 91, 92, 93, 94, 97, 98, 99, 100, 104, 112, 113
Offset: 1

Views

Author

Hugo Pfoertner, Aug 16 2021

Keywords

Examples

			     Diameter  Covered      R^2 =
     of disk   grid        (D/2)^2 =
   n    D      points  A346994(n)/A346995(n)
.
   1 0.00000     1           0   /    1
   2 1.00000     2           1   /    4
   3 1.41421     4           1   /    2
   4 2.00000     5           1   /    1
   5 2.23607     6           5   /    4
   6 2.50000     7          25   /   16
   7 2.82843     9           2   /    1
   8 3.16228    12           5   /    2
   9 3.67696    13         169   /   50
  10 3.80058    14          65   /   18
  11 4.12311    16          17   /    4
  12 4.33333    17         169   /   36
  13 4.47214    21           5   /    1

Links

Crossrefs

The corresponding squared radii are A346994/A346995.
Cf. A053411, A053414, A053415, A123690, A346124.

A346994 Numerators of the squared radii corresponding to circular disks covering record numbers of grid points A346993 of the square lattice.

Original entry on oeis.org

0, 1, 1, 1, 5, 25, 2, 5, 169, 65, 17, 169, 5, 25, 13, 29, 5525, 125, 17, 481, 10, 45, 205, 19721, 1189, 25, 13, 338, 29725, 697, 29, 65, 17, 1105, 3445, 18, 4453, 40885, 4625, 481, 20, 85, 12505, 2125, 200, 89, 45, 7921, 425, 89725, 93925, 2405, 2465, 10201, 98345
Offset: 1

Views

Author

Hugo Pfoertner, Aug 16 2021

Keywords

Examples

			0/1, 1/4, 1/2, 1/1, 5/4, 25/16, 2/1, 5/2, 169/50, 65/18, 17/4, 169/36, ...
For detailed examples see A346993 and the linked pdf.

Links

Crossrefs

The corresponding denominators are A346995.
Cf. A346993, A123690, A346124, A346784, A346785.
All terms of a(n)/A346995(n) with the sole exception of 1/4 are terms of A192493/A192494.

A346995 Denominators of the squared radii corresponding to circular disks covering record numbers of grid points A346993 of the square lattice.

Original entry on oeis.org

1, 4, 2, 1, 4, 16, 1, 2, 50, 18, 4, 36, 1, 4, 2, 4, 722, 16, 2, 50, 1, 4, 18, 1682, 100, 2, 1, 25, 2178, 50, 2, 4, 1, 64, 196, 1, 242, 2178, 242, 25, 1, 4, 578, 98, 9, 4, 2, 338, 18, 3698, 3844, 98, 98, 400, 3844
Offset: 1

Views

Author

Hugo Pfoertner, Aug 16 2021

Keywords

Examples

			0/1, 1/4, 1/2, 1/1, 5/4, 25/16, 2/1, 5/2, 169/50, 65/18, 17/4, 169/36, ...
For detailed examples see A346993 and the linked pdf.

Links

Crossrefs

The corresponding numerators are A346994.
Cf. A346993, A123690, A346124, A346784, A346785.

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).
Showing 1-5 of 5 results.

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