A346123 - OEIS

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.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The segments of the walk can make relative turns of +- 60 degrees. The walks may be open or closed.

LINKS

Table of n, a(n) for n=1..30.

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

Cf. A122223, A127399, A127400, A127401, A258206, A266925.

Cf. A346124-A346132 similar to this sequence with other sets of turning angles.

Sequence in context: A186888 A179883 A179303 * A030309 A074223 A029459

Adjacent sequences: A346120 A346121 A346122 * A346124 A346125 A346126

KEYWORD

nonn,more

AUTHOR

Hugo Pfoertner, Jul 05 2021

STATUS

approved