A256941 -id:A256941 - cp's OEIS frontend

A256940 a(n) is the total number of free ends of a certain configuration of line segments after n iterations (see Comments lines for definition).

2, 4, 8, 12, 12, 12, 20, 20, 16, 24, 28, 48, 52, 36, 44, 36, 16, 24, 40, 56, 72, 72, 76, 80, 60, 64, 80, 124, 132, 88, 100, 68, 16, 24, 40, 56, 72, 80, 88, 104, 112, 128, 176, 216, 244, 212, 168, 148, 84, 64, 104, 152, 200, 200, 212, 216, 148, 144, 176, 276, 296, 192, 212, 136, 16

Offset: 0

Kival Ngaokrajang, Apr 19 2015

nonn

The initial pattern is a straight line segment which has 2 free ends: a(0)=2.
The construction rules for the following generations are:
(i) add 2 line segments (all line segments are of equal length) at each free end of previous generation by arranging them in a "V" shape at angle Pi/2 and symmetrically placed at the free end,
(ii) overlaps among different generations are prohibited (if, for a given free end, any of the two new segments from its "V" touch or cross a segment from an earlier generation, then the entire "V" is not added, and that free end is just declared non-free),
(iii) the {a(n)} free ends are the ends of elements that do not touch or cross the others (if a new segment is touched by another segment only at the endpoint which it shares with its parent, then this doesn't count as an intersection and its other end is considered free).
It seems that a(n) drop to 16 for n = 8, 16, 32, 64,... . See illustration in the links.
The structure of the illustration of initial terms is very similar to the structure of A194270 and A220500. - Omar E. Pol, Apr 19 2015

Kival Ngaokrajang, Illustration of initial terms, n <= 12
Andrey Zabolotskiy, Illustration for a(64)=16

Cf. A194270, A194440, A194442, A220500, A220520, A220522, A256641, A256941.

new2[{{s_, t_}, a_}] := Simplify@Table[{{t, AngleVector[t, {1, a + si Pi/4}]}, a + si Pi/4}, {si, {1, -1}}];
xx[l1_, l2_] := SquaredEuclideanDistance[First@l1, First@l2] <= 4 && With[{int = Simplify@RegionIntersection[Line@l1, Line@l2]}, int =!= EmptyRegion[2] && int =!= Point[{First@l2}] && int =!= Point[{First@l1}]];
{nonfree, free} = {{}, {{{{1/2, 0}, {1, 0}}, 0}, {{{1/2, 0}, {0, 0}}, Pi}}};
a = {2};
next[] := ({oldnonfree, oldfree, nonfree, free} = {nonfree, free, Join[free, nonfree], {}};
  Do[n2 = new2[f]; If[And @@ Table[AllTrue[oldnonfree, ! xx[First@#, First@new] &], {new, n2}], Do[
    tt = GroupBy[free, xx[First@#, First@new] &];
    free = Lookup[tt, False, {}];
    If[KeyExistsQ[tt, True], nonfree = Join[nonfree, tt[True], {new}], AppendTo[free, new]];
  , {new, n2}]], {f, oldfree}];
  AppendTo[a, Length@free];);
Do[next[], {10}];
a (* Andrey Zabolotskiy, Mar 09 2025 *)

a(1) = 2 prepended and a(3) = 8 corrected by Omar E. Pol, Apr 19 2015
Partially edited by Kival Ngaokrajang, as Omar E. Pol suggestion, Apr 26 2015
Terms a(12), a(13), a(59) corrected by Kival Ngaokrajang, Apr 26 2015
Terms a(27), a(60), a(63) corrected, other terms verified, description clarified by Andrey Zabolotskiy, Mar 09 2025

cp's OEIS Frontend

A256940 a(n) is the total number of free ends of a certain configuration of line segments after n iterations (see Comments lines for definition).

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