OFFSET
1,2
COMMENTS
The segments of the walk can make relative turns of +- 60 degrees. The walks may be open or closed.
LINKS
Hugo Pfoertner, Examples of paths of maximum length.
FORMULA
a(n+1) >= a(n) + 1 for n > 1; a(1) = 1.
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Hugo Pfoertner, Jul 05 2021
STATUS
approved