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A346123
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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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10
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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
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OFFSET
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1,2
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COMMENTS
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The segments of the walk can make relative turns of +- 60 degrees. The walks may be open or closed.
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LINKS
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FORMULA
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a(n+1) >= a(n) + 1 for n > 1; a(1) = 1.
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EXAMPLE
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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 .%
% \ / % % \ / %
% \ / % % \ / %
% \ / % % \ / %
% \ / % % \ / %
% ---------------- % % ---------------- %
%%% %%%% %%% %%% %%%% %%%
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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.
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CROSSREFS
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Cf. A346124-A346132 similar to this sequence with other sets of turning angles.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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