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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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).

Original entry on oeis.org

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

Views

Author

Kival Ngaokrajang, Apr 19 2015

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

A256941 a(n) is the number of free ends of a certain configuration of line segments after n iterations (see Comments lines for definition).

Original entry on oeis.org

2, 4, 8, 12, 16, 24, 28, 32, 32, 24, 32, 48, 60, 64, 68, 72, 48, 24, 32, 56, 88, 120, 120, 120, 104, 76, 80, 120, 140, 144, 148, 152, 80, 24, 32, 56, 88, 128, 168, 224, 256, 256, 212, 216, 232, 244, 224, 240, 188, 92, 80, 144, 232, 296, 296, 296, 256, 180, 176, 264, 300, 304, 308, 312, 144, 24
Offset: 0

Views

Author

Kival Ngaokrajang, Apr 19 2015

Keywords

Comments

All line segments are equal length. The initial pattern is a straight line segment. It has 2 free ends, so a(0)=2. The construction rules for generation n >= 1 are:
(i) subject to rule ii, add 2 line segments at each free end by arranging in a "V" shape with angle Pi/3 and connecting symmetrically to the free end (nearly like a 3-handed clock showing 07:00:25);
(ii) a "V" is not added if either of its segments would cross a line segment drawn in an earlier generation;
(iii) when generation n is complete, each new line segment clearly touches 2 line segments where it was initially attached; the other end of the new line segment counts as being free if the segment does not touch or cross any more line segments.
a(n) is the number of free ends created in generation n.
It seems that a(n) drops to 24 for n = 5, 9, 17, 33, 65, ... . See illustrations in the links.
The terms of this sequence should be checked! - Omar E. Pol, Apr 23 2015

Links

Crossrefs

Cf. A194270, A194440, A194442, A220500, A220520, A220522, A256641, A256940.

Extensions

First term suggested by Omar E. Pol, Apr 23 2015
Author's comments edited by Peter Munn, May 11 2021

A230978 a(n) = (A220500(n) - 1)/2.

Original entry on oeis.org

0, 2, 6, 14, 25, 37, 49, 67, 87, 103, 125, 157, 204, 240, 268, 306, 342, 358, 382, 422, 478, 548, 618, 688, 772, 832, 898, 982, 1101, 1185, 1245, 1323, 1391, 1407, 1431, 1471, 1527, 1597, 1669, 1751, 1863, 1971, 2099, 2235, 2419, 2581, 2739, 2879, 3027, 3107, 3182, 3298, 3458, 3660, 3876, 4080
Offset: 1

Views

Author

Omar E. Pol, Nov 30 2013

Keywords

Links

Crossrefs

Cf. A139250, A220500, A220520, A220522, A220526.

Extensions

More terms added using the data at A220500 by Amiram Eldar, May 06 2024

A231346 Number of distinct polygonal shapes after n-th stage in the structure of the D-toothpick cellular automaton of A220500.

Original entry on oeis.org

0, 0, 0, 1, 3, 4, 5, 7, 8, 8, 8, 11, 15, 17, 18, 19, 19, 19, 19, 19, 22
Offset: 1

Views

Author

Omar E. Pol, Dec 07 2013

Keywords

Comments

The cellular automaton of A220500 contains a large number of distinct polygonal shapes. The exact number is unknown. Apparently it's greater than 63.
For simplicity we also call polygonal shapes "polygons".
In order to construct this sequence we use the following rules:
- Consider only the convex polygons and the concave polygons. Self-intersecting polygons are not counted.
- Unfinished polygons with inward growth are not counted.
- If two polygons have the same shape but they have different size then these polygons must be counted as distinct polygonal shapes.
- The reflected shapes of asymmetric polygons, both with the same area, must be counted as distinct polygonal shapes.
Question: Is there a maximal record in this sequence?

Links

Crossrefs

Cf. A139250, A194276, A220500, A220512, A220514, A220520, A220522, A220524, A220526.
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