A220500 -id:A220500 - cp's OEIS frontend

A323646 "Letter A" toothpick sequence (see Comments for precise definition).

0, 1, 3, 5, 9, 15, 21, 27, 39, 53, 65, 71, 83, 97, 113, 131, 163, 197, 217, 223, 235, 249, 265, 283, 315, 349, 373, 391, 423, 461, 505, 567, 659, 741, 777, 783, 795, 809, 825, 843, 875, 909, 933, 951, 983, 1021, 1065, 1127, 1219, 1301, 1341, 1359, 1391, 1429, 1473, 1535, 1627, 1713, 1773, 1835, 1931

Offset: 0

Omar E. Pol, Mar 07 2019

nonn

This arises from a hybrid cellular automaton formed of toothpicks of length 2 and D-toothpicks of length 2*sqrt(2).
For the construction of the sequence the rules are as follows:
On the infinite square grid at stage 0 there are no toothpicks, so a(0) = 0.
For the next n generations we have that:
At stage 1 we place a toothpick of length 2 in the horizontal direction, centered at [0,0], so a(1) = 1.
If n is even we add D-toothpicks. Each new D-toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.
If the x-coordinate of the middle point of the D-toothpick is negative then the D-toothpick must be placed in the NE-SW direction.
If the x-coordinate of the middle point of the D-toothpick is positive then the D-toothpick must be placed in the NW-SE direction.
If n is odd we add toothpicks in horizontal direction. Each new toothpick must have its midpoint touching the endpoint of exactly one existing D-toothpick.
The sequence gives the number of toothpicks and D-toothpicks after n stages.
A323647 (the first differences) gives the number of elements added at the n-th stage.
Note that if n >> 1 at the end of every cycle the structure looks like a "volcano", or in other words, the structure looks like a trapeze which is almost an isosceles right triangle.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

			After two generations the structure looks like a letter "A" which is formed by a initial I-toothpick (or a toothpick of length 2), placed in horizontal direction, and two D-toothpicks each of length 2*sqrt(2) as shown below, so a(2) = 3.
Note that angle between both D-toothpicks is 90 degrees.
.
                      *
                    *   *
                  * * * * *
                *           *
              *               *
.
After three generations the structure contains three horizontal toothpicks and two D-toothpicks as shown below, so a(3) = 5.
.
                      *
                    *   *
                  * * * * *
                *           *
          * * * * *       * * * * *
.

Cf. A139250, A168112, A160730, A296612, A323647.

For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A292612, A299770, A323650, A327330, A327332.

a(n) = 1 + A160730(n-1), n >= 1.

a(n) = 1 + 2*A168112(n-1), n >= 1.

A327330 "Concave pentagon" toothpick sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 33, 41, 45, 53, 63, 75, 89, 111, 133, 149, 153, 161, 171, 183, 197, 219, 241, 261, 275, 299, 327, 361, 403, 463, 511, 547, 551, 559, 569, 581, 595, 617, 639, 659, 673, 697, 725, 759, 801, 861, 909, 949, 967, 995, 1029, 1075, 1125, 1183, 1233, 1281, 1321, 1389, 1465, 1549, 1657

Offset: 0

Omar E. Pol, Sep 01 2019

nonn

This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks (A160164) and V-toothpicks (A161206).
The surprising fact is that after 2^k stages the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two quadrilaterals (Q1 and Q2), both with their largest sides in vertical position, as shown below:
.
                  *                 *
                  *  *           *  *
                  *     *     *     *
                  *        *        *
                  *   Q1   *   Q2   *
                  *       * *       *
                  *      *   *      *
                  *     *     *     *
                  *    *       *    *
                  *   *    E    *   *
                  *  *           *  *
                  * *             * *
                  **               **
                  * * * * * * * * * *
.
Note that for n >> 1 both quadrilaterals look like right triangles.
Every polygon has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.
For the construction of the sequence the rules are as follows:
On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
At stage 1 we place an I-toothpick formed of two single toothpicks in vertical position, so a(1) = 1.
For the next n generation we have that:
If n is even then at every free end of the structure we add a V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof.
If n is odd then we add I-toothpicks in vertical position (see the example).
a(n) gives the total number of I-toothpicks and V-toothpicks in the structure after the n-th stage.
A327331 (the first differences) gives the number of elements added at the n-th stage.
2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12-gons, concave 18-gons, concave 20-gons, and other polygons.
The structure is almost identical to the structure of A327332, but a little larger at the upper edge.
The behavior seems to suggest that this sequence can be calculated with a formula, in the same way as A139250, but that is only a conjecture.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
For another version, very similar, starting with a V-toothpick, see A327332, which it appears that shares infinitely many terms with this sequence.

			Illustration of initial terms:
.
               |      /|\     |/|\|
               |       |      | | |
                      / \     |/ \|
                              |   |
n   :  0       1       2        3
a(n):  0       1       3        7
After three generations there are five I-toothpicks and two V-toothpicks in the structure, so a(3) = 5 + 2 = 7 (note that in total there are 2*a(3) = 2*7 = 14 single toothpicks of length 1).

First differs from A231348 at a(11).
Cf. A047999, A139250 (normal toothpicks), A160164 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), A296612, A323641, A323642, A327331 (first differences), A327332 (another version).
For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.

Conjecture: a(2^k) = A327332(2^k), k >= 0.

A327332 "Concave pentagon" toothpick sequence, starting with a V-toothpick (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 21, 33, 41, 45, 51, 63, 75, 85, 101, 133, 149, 153, 159, 171, 183, 193, 209, 241, 261, 273, 291, 327, 363, 389, 431, 515, 547, 551, 557, 569, 581, 591, 607, 639, 659, 671, 689, 725, 761, 787, 829, 913, 953, 969, 993, 1041, 1085, 1109, 1149, 1229, 1277, 1309, 1357, 1453, 1549, 1613

Offset: 0

Omar E. Pol, Sep 01 2019

nonn

Another version and very similar to A327330.
This arises from a hybrid cellular automaton on a triangular grid formed of V-toothpicks (A161206) and I-toothpicks (A160164).
After 2^k stages, the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two right triangles (R1 and R2) both with their hypotenuses in vertical position, as shown below:
.
                  *                 *
                  *  *           *  *
                  *     *     *     *
                  *        *        *
                  *  R1   * *   R2  *
                  *      *   *      *
                  *     *     *     *
                  *    *       *    *
                  *   *    E    *   *
                  *  *           *  *
                  * *             * *
                  **               **
                  * * * * * * * * * *
.
Every triangle has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.
For the construction of the sequence the rules are as follows:
On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
At stage 1 we place an V-toothpick, formed of two single toothpicks, with its central vertice directed up, like a gable roof, so a(1) = 1.
For the next n generation we have that:
If n is even then at every free end of the structure we add a I-toothpick formed of two single toothpicks in vertical position.
If n is odd then at every free end of the structure we add a V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof (see the example).
a(n) gives the total number of V-toothpicks and I-toothpicks in the structure after the n-th stage.
A327333 (the first differences) gives the number of elements added at the n-th stage.
2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12-gons, concave 18-gons, concave 20-gons, and other polygons.
The structure is almost identical to the structure of A327330, but a little smaller.
The behavior seems to suggest that this sequence can be calculated with a formula, in the same way as A139250, but that is only a conjecture.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
It appears that A327330 shares infinitely many terms with this sequence.

			Illustration of initial terms:
.
.             /\     |/\|
.                    |  |
.
n:     0       1       2
a(n):  0       1       3
After two generations there are only one V-toothpick and two I-toothpicks in the structure, so a(2) = 1 + 2 = 3 (note that in total there are 2*a(2)= 2*3 = 6 single toothpicks of length 1).

Cf. A139250 (normal toothpicks), A160164 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), A296612, A323641, A323642, A327333 (first differences), A327330 (another version).

For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.

Conjecture: a(2^k) = A327330(2^k), k >= 0.

A220512 D-toothpick sequence of the third kind starting with a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 12, 28, 44, 60, 88, 136, 168, 184, 216, 280, 344, 424, 508, 620, 684, 700, 732, 796, 892, 1004, 1148, 1324, 1460, 1572, 1668, 1844, 2020, 2228, 2424, 2664, 2792, 2808, 2840, 2904, 3000, 3112, 3264

Offset: 0

Omar E. Pol, Dec 15 2012

This is a toothpick sequence of forking paths to 135 degrees. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A221565 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure has a fractal (or fractal-like) behavior. For more information see A194700.

First differs from A194432 at a(14).

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
turtle graphics, D-Toothpick Sequence, Youtube video (2023)
Index entries for sequences related to toothpick sequences

Cf. A139250, A194270, A194432, A194700, A220500, A220514, A220520, A220522, A220524, A220526.

A233760 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "wide" triangle of the third kind, second version.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 53, 67, 79, 93, 109, 111, 115, 121, 133, 147, 167, 189, 215, 237, 249, 267, 299, 337, 365, 395, 427, 429, 433, 439, 451, 465, 485, 507, 533, 555, 575, 605, 649, 711, 763, 809, 863, 901, 913, 931, 967, 1013

Offset: 0

Omar E. Pol, Dec 16 2013

nonn

The structure is essentially the same as A220520 but here the borders do not contain toothpicks with exposed endpoints except the initial toothpick. The structure is one of the oblique wedges of several D-toothpick structures. For more information see A220500. The first differences (A233761) give the number of toothpicks or D-toothpicks added at n-th stage.

Cf. A139250, A220500, A220520, A233761, A233762, A233780.

A233764 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 201, 209, 223, 245, 271, 287, 317, 349, 351, 355, 361, 371, 381, 391, 409, 433, 451, 461, 479, 507, 545, 575, 625, 679, 713, 721, 735, 757, 783

Offset: 0

Omar E. Pol, Dec 16 2013

nonn

The D-toothpicks placed in northwest or northeast direction both are prohibited. Note that due this rule there are substructures with broken symmetry, for instance a(44) = 507, not 509. For another version without broken symmetry see A233780.
A233765 (the first differences) gives the number of toothpicks or D-toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from A233970 at a(25).
First differs from A233780 at a(44).

Cf. A139250, A169780, A220500, A220520, A231348, A233765, A233780, A233970.

A233762 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the third kind, second version.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 20, 28, 30, 34, 40, 52, 58, 64, 75, 91, 93, 97, 103, 115, 129, 145, 163, 187, 197, 205, 219, 247, 261, 277, 300, 332, 334, 338, 344, 356, 370, 386, 406, 438, 464, 484, 506, 546, 576, 616, 655, 703, 721, 729

Offset: 0

Omar E. Pol, Dec 15 2013

nonn

The structure is essentially the same as A220522 but here the borders do not contain D-toothpicks with exposed endpoints except the initial D-toothpick. The structure is one of the oblique wedges of several D-toothpick structures. For more information see A220500. The first differences (A233763) give the number of toothpicks or D-toothpicks added at n-th stage.

Cf. A139250, A220500, A220522, A233760, A233763.

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

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

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

Kival Ngaokrajang, Apr 19 2015

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

Kival Ngaokrajang, Illustration of initial terms, n <= 11
Kival Ngaokrajang, Illustration for a(65) = 24

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

First term suggested by Omar E. Pol, Apr 23 2015

Author's comments edited by Peter Munn, May 11 2021

cp's OEIS Frontend

A323646 "Letter A" toothpick sequence (see Comments for precise definition).

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A327330 "Concave pentagon" toothpick sequence (see Comments for precise definition).

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A327332 "Concave pentagon" toothpick sequence, starting with a V-toothpick (see Comments for precise definition).

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A220512 D-toothpick sequence of the third kind starting with a cross formed by 4 toothpicks.

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A233760 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "wide" triangle of the third kind, second version.

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A233764 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

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A233762 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the third kind, second version.

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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).

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

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