A266925 -id:A266925 - cp's OEIS frontend

A151541 Number of 2-sided triangular strip polyedges with n cells.

1, 3, 8, 32, 123, 523, 2201, 9443, 40341, 172649, 736926, 3141607, 13367012, 56790498, 240919918, 1020753475, 4319803799, 18262494912, 77134873774, 325518862387, 1372679840360, 5784417772262

Offset: 1

Ed Pegg Jr, May 13 2009

Also number of unrooted self-avoiding walks of n steps on hexagonal [ =triangular ] lattice. - Hugo Pfoertner, Jun 23 2018

Ed Pegg, Jr., Illustrations of polyforms
Hugo Pfoertner, Illustration of ratio A001334(n)/a(n) using Plot2.
Eric Weisstein's World of Mathematics, Polyedge

Cf. A037245, A151539, A151540, A159867, A266925.

Asymptotically approaches (1/24)*A001334(n) for increasing n.

a(9)-a(13) from Joseph Myers, Oct 05 2011

a(14)-a(22) from Bert Dobbelaere, Mar 23 2025

A306178 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/4, +-Pi/2, +-3*Pi/4.

Original entry on oeis.org

1, 4, 15, 86, 492, 2992, 18172, 110643, 672267, 4069122, 24578785, 147972210, 889332713, 5331980703, 31924424199

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular octagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306179, A306180, A306181, A306182.

a(7)-a(15) from Bert Dobbelaere, May 15 2025

A306175 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/5, +-3*Pi/5.

Original entry on oeis.org

1, 2, 6, 19, 66, 229, 831, 2991, 10859, 39173, 141631, 510079, 1835583, 6586932, 23614821, 84492315, 302014619, 1077860479, 3843695976, 13690026688

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular pentagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 5.

Cf. A266925, A037245, A151541, A306177, A306178, A306179, A306180, A306181, A306182, A316195.

a(7)-a(9) from Hugo Pfoertner, Dec 23 2018

a(10)-a(20) from Bert Dobbelaere, May 15 2025

A306177 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/7, +-3Pi/7, +-5Pi/7.

Original entry on oeis.org

1, 3, 11, 55, 275, 1444, 7609, 40220, 212051, 1114206, 5840287, 30521627, 159166728, 828130291, 4301636648

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular heptagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 4.

Cf. A266925, A037245, A306175, A151541, A306178, A306179, A306180, A306181, A306182.

a(7)-a(15) from Bert Dobbelaere, May 15 2025

A306179 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/9, +-3Pi/9, +-5Pi/9, +-7*Pi/9.

Original entry on oeis.org

1, 4, 19, 128, 861, 6051, 42475, 297993, 2081607, 14485077, 100475175, 694838429, 4793805002

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular nonagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306180, A306181, A306182.

a(7)-a(13) from Bert Dobbelaere, May 15 2025

A306180 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/5, +-2Pi/5, +-3Pi/5, +-4*Pi/5.

Original entry on oeis.org

1, 5, 23, 169, 1233, 9551

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular decagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306179, A306181, A306182.

A306181 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/11, +-3Pi/11, +-5Pi/11, +-7Pi/11, +-9Pi/11.

Original entry on oeis.org

1, 5, 28, 235, 1970, 17201, 149420, 1295637, 11178026, 96047288, 822418731, 7020762655

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular 11-gon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306179, A306180, A306182.

a(7)-a(12) from Bert Dobbelaere, May 15 2025

A306182 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/6, +-2Pi/6, +-3Pi/6, +-4Pi/6, +-5Pi/6.

Original entry on oeis.org

1, 6, 33, 306, 2765, 26290, 247737, 2332965, 21856232, 204019196, 1897940592, 17606864337

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular 12-gon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306179, A306180, A306181.

a(7)-a(12) from Bert Dobbelaere, May 15 2025

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.

cp's OEIS Frontend

A151541 Number of 2-sided triangular strip polyedges with n cells.

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A306178 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/4, +-Pi/2, +-3*Pi/4.

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A306175 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/5, +-3*Pi/5.

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A306177 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/7, +-3*Pi/7, +-5*Pi/7.

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A306179 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/9, +-3*Pi/9, +-5*Pi/9, +-7*Pi/9.

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A306180 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/5, +-2*Pi/5, +-3*Pi/5, +-4*Pi/5.

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A306181 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/11, +-3*Pi/11, +-5*Pi/11, +-7*Pi/11, +-9*Pi/11.

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A306182 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/6, +-2*Pi/6, +-3*Pi/6, +-4*Pi/6, +-5*Pi/6.

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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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A306177 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/7, +-3Pi/7, +-5Pi/7.

A306179 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/9, +-3Pi/9, +-5Pi/9, +-7*Pi/9.

A306180 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/5, +-2Pi/5, +-3Pi/5, +-4*Pi/5.

A306181 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/11, +-3Pi/11, +-5Pi/11, +-7Pi/11, +-9Pi/11.

A306182 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/6, +-2Pi/6, +-3Pi/6, +-4Pi/6, +-5Pi/6.