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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.
10
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
1,2
COMMENTS
The segments of the walk can make relative turns of +- 60 degrees. The walks may be open or closed.
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
Cf. A346124-A346132 similar to this sequence with other sets of turning angles.
Sequence in context: A186888 A179883 A179303 * A030309 A074223 A029459
KEYWORD
nonn,more
AUTHOR
Hugo Pfoertner, Jul 05 2021
STATUS
approved