A194434 -id:A194434 - cp's OEIS frontend

A194435 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194434.

0, 4, 8, 16, 16, 16, 32, 44, 32, 16, 32, 64, 96, 48, 80, 100, 64, 16, 32, 64, 96, 112, 144, 168, 176, 80, 96, 160, 256, 128, 176, 212, 128, 16, 32, 64, 96, 112, 144, 176, 208, 168, 192, 240, 400, 272, 336, 332, 336, 112, 96, 176, 288, 336, 416, 464

Offset: 0

Omar E. Pol, Sep 03 2011

Essentially the first differences of A194434.
First differs from A221528 at a(13). - Omar E. Pol, Mar 23 2013
From Omar E. Pol, Jun 24 2022: (Start)
The word of this cellular automaton is "ab".
For the nonzero terms the structure of the irregular triangle is as shown below:
  a,b;
  a,b;
  a,b,a,b;
  a,b,a,b,a,b,a,b;
  a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
  ...
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
Columns "a" contain numbers of D-toothpicks (of length sqrt(2)).
Columns "b" contain numbers of toothpicks (of length 1).
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
For further information about the word of cellular automata see A296612. (End)

			From _Omar E. Pol_, Mar 23 2013: (Start)
When written as an irregular triangle the sequence of nonzeros terms begins:
   4, 8;
  16,16;
  16,32,44,32;
  16,32,64,96, 48, 80,100, 64;
  16,32,64,96,112,144,168,176, 80, 96,160,256,128,176,212,128;
  16,32,64,96,112,144,176,208,168,192,240,400,272,336,332,336,112,96, ...
  ... (End)
Right border gives the powers of 2 >= 8 (reformatted the triangle). - _Omar E. Pol_, Jun 24 2022

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Paolo Xausa, Animated version for n = 0..31 (red elements)
Paolo Xausa, Animated version for n = 0..63 (red elements)
Paolo Xausa, Illustration of initial terms for n = 0..63 (red elements, multipage PDF)
Index entries for sequences related to toothpick sequences

Cf. A011782, A139251, A194271, A194433, A194434, A194441, A194443, A194445, A212009, A221528, A299612.

a(n) = 4*A194445(n).

Conjecture: a(2^k+1) = 16, if k >= 1.

More terms from Omar E. Pol, Mar 23 2013

A194270 D-toothpick sequence of the second kind (see Comments lines for definition).

Original entry on oeis.org

0, 1, 5, 13, 29, 51, 75, 97, 137, 177, 209, 241, 297, 371, 467, 517, 605, 677, 709, 757, 829, 933, 1061, 1173, 1317, 1461, 1613, 1709, 1861, 2039, 2279, 2401, 2585, 2721, 2753, 2801, 2873, 2981, 3125, 3269, 3453, 3641, 3841, 4017, 4289, 4563, 4979, 5229

Offset: 0

Omar E. Pol, Aug 23 2011

nonn

This is a cellular automaton of forking paths to 135 degrees which uses elements of two sizes: toothpicks of length 1 and D-toothpicks of length 2^(1/2). Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints.
On the infinite square grid we start with no elements.
At stage 1, we place anywhere a D-toothpick.
The rule for adding new elements is as follows. Each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees. Intersections and overlapping are prohibited, so some toothpick endpoints can remain exposed forever.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194271) give the number of toothpicks and D-toothpicks added at n-th stage.
If n >> 1 the structure looks like an almost regular octagon. The structure has a fractal-like behavior related to powers of 2 (see formula section in A194271 and A194443). Note that for some values of n we can see an internal growth, similar to A160120. Also there are hidden substructures which have a surprising connection with the Sierpinski triangle. The hidden substructures are displayed more clearly for large values of n without reducing the scale of the drawing. The main "wedges" in the structures are essentially the triangles A194440 and A194442.
Note that this structure is much more complex than A139250 and A160120. The structure contains a large number of distinct polygonal shapes. There are convex polygons and concave polygons, also there are symmetrical and asymmetrical polygons. Several of these polygons are also in the structure of A172310. The number of edges of the known polygons are 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 20, 24. Is not known how many distinct types of polygons there are in the structure if n -> infinite. The sequences related with these polygons are A194276, A194277, A194278 and A194283. Note that the structure is not centered with respect to the axes X, Y. Also, for some polygons the area is not an integer. For symmetric versions of C. A. see A194432 and A194434.
Another representation (Large version): instead toothpicks of length 1 we place toothpicks of length 2. We start with no toothpicks. At stage 1, we place a toothpick of length 2 on the y-axis and centered at the origin. At stage 2 we place four D-toothpicks of length 2^(1/2) = sqrt(2), and so on. In this case the structure is centered with respect to the axes X, Y and the area of the polygons is an integer.
[It appears that a normal toothpick is a line segment of length 1 that is parallel to the x-axis or the y-axis. A D-toothpick is a line segment of length sqrt(2) with slope +-1. D stands for diagonal. - N. J. A. Sloane, Feb 06 2023]

			Illustration of initial terms:
.                                            o   o
.                                             \ /
.                        o                     o     o
.                        |                     |    /
.        o               o - o         o       o - o
.       /               /               \     /     \
.      o           o - o                 o - o       o
.                      |                /    |
.                      o               o     o
.                                           / \
.                                          o   o
.
.       1               5                    13
.

David Applegate, The movie version
Ayliean McDonald, D-toothpick fractal, TikTok video.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A160120, A172310, A182838, A194271, A194276, A194277, A194278, A194440, A194441, A194442, A194443, A194444.

More terms from Omar E. Pol, Sep 01 2011

A220500 D-toothpick sequence of the third kind starting with a single toothpick.

Original entry on oeis.org

0, 1, 5, 13, 29, 51, 75, 99, 135, 175, 207, 251, 315, 409, 481, 537, 613, 685, 717, 765, 845, 957, 1097, 1237, 1377, 1545, 1665, 1797, 1965, 2203, 2371, 2491, 2647, 2783, 2815, 2863, 2943, 3055, 3195, 3339, 3503, 3727, 3943, 4199, 4471, 4839, 5163, 5479, 5759, 6055, 6215, 6365, 6597, 6917, 7321, 7753, 8161

Offset: 0

Omar E. Pol, Dec 15 2012

nonn

This is a cellular automaton of forking paths to 135 degrees which uses elements of three sizes: toothpicks of length 1, D-toothpicks of length 2^(1/2) and D-toothpicks of length 2^(1/2)/2. Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints.
On the infinite square grid we start with no elements.
At stage 1, place a single toothpick on the paper, aligned with the y-axis. The rule for adding new elements is as follows. Each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees. Intersections and overlapping are prohibited.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A220501) give the number of toothpicks or D-toothpicks added at n-th stage.
It appears that if n >> 1 the structure looks like an octagon. This C.A. has a fractal (or fractal-like) behavior related to powers of 2. Note that for some values of n we can see an internal growth.
The structure contains eight wedges. Each vertical wedge (see A220520) also contains infinitely many copies of the oblique wedges. Each oblique wedge (see A220522) also contains infinitely many copies of the vertical wedges. Finally, each horizontal wedge also contains infinitely many copies of the vertical wedges and of the oblique wedges.
The structure is mysterious: it contains at least 59 distinct internal regions (or polygonal pieces), for example: one of the concave octagons appears for first time at stage 223. The largest known polygon is a concave 24-gon. The exact number of distinct polygons is unknown.
Also the structure contains infinitely many copies of two subsets of distinct size which are formed by five polygons: three hexagons, a 9-gon and a pentagon. These subsets have a surprising connection with the Sierpinski triangle A047999, but the pattern is more complex.
Apparently this cellular automaton has the most complex structure of all the toothpick structures that have been studied (see illustrationsm also the illustrations of the wedges in the entries A220520 and A220522).
The structure contains at least 69 distinct polygonal pieces. The largest known polygon is a concave 24-gon of area 95/2 = 47.5 which appears for first time at stage 879. - Omar E. Pol, Feb 10 2018

David Applegate, The movie version
Leonid Broukhis, A program generating variations of the "D-toothpick" pattern, Oct. 1995; (remarks)
Ayliean McDonald, 2 Minute Mathematical Meditation, Youtube video (2021).
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the structure after 17 stages
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A172310, A194270, A194276, A194432, A194434, A194440, A194442, A194444, A194700, A220520, A220522.

Terms a(23) and beyond from David Applegate's movie version. - Omar E. Pol, Feb 10 2018

A194444 D-toothpick sequence of the second kind in the first quadrant.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 34, 42, 46, 54, 70, 94, 106, 126, 151, 167, 171, 179, 195, 219, 247, 283, 325, 369, 389, 413, 453, 517, 549, 593, 646, 678, 682, 690, 706, 730, 758, 794, 838, 890, 932, 980, 1040, 1140, 1208, 1292, 1375, 1459, 1487, 1511, 1555

Offset: 0

Omar E. Pol, Aug 24 2011

nonn

This cellular automaton has essentially the same rules as A194270. We start at stage 0 with no toothpicks. At stage 1, we place a D-toothpick of length sqrt(2), in diagonal direction, at (0,0),(1,1). At stage 2, we place two toothpicks of length 1. At stage 3 we place four D-toothpicks. And so on. The toothpicks and D-toothpicks are connected by their endpoints. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194445) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure shows a fractal (or fractal-like) behavior.

First differs from A220524 at a(13). - Omar E. Pol, Mar 23 2013

Cf. A139250, A172310, A182838, A194270, A194432, A194434, A194440, A194442, A194445, A212008, A220524.

a(n) = A194434(n)/4. - Omar E. Pol, Oct 15 2011

More terms from Omar E. Pol, Mar 23 2013

A160740 Toothpick sequence starting from a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 8, 16, 24, 32, 40, 56, 72, 80, 88, 104, 120, 136, 160, 200, 232, 240, 248, 264, 280, 296, 320, 360, 392, 408, 432, 472, 512, 560, 640, 744, 808, 816, 824, 840, 856, 872, 896, 936, 968, 984, 1008, 1048, 1088, 1136, 1216, 1320, 1384, 1400, 1424, 1464, 1504, 1552

Offset: 0

Omar E. Pol, May 25 2009

nonn

On the infinite square grid we start at stage 0 with no toothpicks. Toothpicks have length 2. At stage 1 we place two consecutive toothpicks in the vertical direction and two consecutive toothpicks in the horizontal direction forming a cross centered at the origin. At stage 2 we place four toothpicks. At stage 3 we place eight toothpicks. For more information about the toothpick sequences see A139250. - Omar E. Pol, Nov 24 2011

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.], which is also available at arXiv:1004.3036v2
Mathematical Association of America, NumberADay: 1136
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A139251, A160736, A160738, A160741.

Cf. A160170, A160426, A194432, A194434.

a(n) = 4*A160406(n).

More terms from N. J. A. Sloane, May 25 2009

A194432 D-toothpick sequence starting with a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 12, 28, 44, 60, 88, 136, 168, 184, 216

Offset: 0

Omar E. Pol, Sep 03 2011

On the infinite square grid we start with no toothpicks.
At stage 1, we place a cross, centered at the origin, formed by 2 vertical toothpicks and 2 horizontal toothpicks of length 1. At stage 2, we place 8 D-toothpicks of length sqrt(2). At stage 3, we place 16 toothpicks of length 1. And so on.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194433) give the number of toothpicks and D-toothpicks added at n-th stage.
Apparently this cellular automaton has a fractal (or fractal-like) behavior related to power of 2, similar to A194270 and very similar to A194434. The octagonal structure contains a large number of distinct polygonal shapes. For more information see A194270, A194440 and A194442.

Cf. A139250, A194270, A194433, A194434, A194440, A194442.

A220514 D-toothpick sequence of the third kind starting with a X-shaped cross formed by 4 D-toothpicks.

Original entry on oeis.org

0, 4, 12, 28, 44, 60, 92, 136, 168, 184, 216, 280, 376, 440, 520, 620, 684, 700, 732, 796, 892, 1020, 1164, 1332, 1508, 1588, 1684, 1860, 2116, 2276, 2452, 2664, 2792, 2808, 2840, 2904, 3000, 3128, 3272, 3448, 3656, 3824, 4016, 4272, 4676, 4992

Offset: 0

Omar E. Pol, Dec 15 2012

nonn

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. A221528 (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 A194434 at a(13).

Cf. A139250, A194270, A194700, A194434, A220500, A220520, A220522, A221528.

a(n) = 4*A220524(n).

A212008 D-toothpick sequence of the second kind starting with a single toothpick.

Original entry on oeis.org

0, 1, 5, 13, 29, 51, 71, 95, 131, 171, 203, 247, 303, 397, 457, 513, 589, 661, 693, 741, 813, 925, 1057, 1197, 1333, 1501, 1613, 1745, 1885, 2123, 2271, 2391, 2547, 2683, 2715, 2763, 2835, 2947, 3079

Offset: 0

Omar E. Pol, Dec 15 2012

This cellular automaton uses elements of two sizes: toothpicks of length 1 and D-toothpicks of length 2^(1/2). Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints.
On the infinite square grid we start with no elements.
At stage 1, place a single toothpick on the paper, aligned with the y-axis.
The rule for adding new elements is as follows. If it is possible, each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees, otherwise each exposed endpoint of the elements of the old generation must be touched by an endpoint of an element of the new generation such that the angle between the old element and the new element is equal to 135 degrees. Intersections and overlapping are prohibited. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A212009) give the number of toothpicks or D-toothpicks added at n-th stage.
It appears that if n >> 1 the structure looks like an octagon. This C.A. has a fractal (or fractal-like) behavior related to powers of 2. Note that for some values of n we can see an internal growth.
The structure contains eight wedges. Each vertical wedge also contains infinitely many copies of the oblique wedges. Each oblique wedge also contains infinitely many copies of the vertical wedges. Finally, each horizontal wedge also contains infinitely many copies of the vertical wedges and of the oblique wedges.
The structure appears to be a puzzle which contains at least 50 distinct internal regions (or polygonal pieces), and possibly more. Some of them appear for first time after 200 stages. The largest known polygon is a concave 24-gon.
Also the structure contains infinitely many copies of two subsets of distinct size which are formed by five polygons: three hexagons, a 9-gon and a pentagon. The distribution of these subsets have a surprising connection with the Sierpinski triangle A047999, but here the pattern is more complex.
For another version see A220500.

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A047999, A139250, A194270, A194432, A194434, A194440, A194442, A194444, A220500.

cp's OEIS Frontend

A194435 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194434.

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A194270 D-toothpick sequence of the second kind (see Comments lines for definition).

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A220500 D-toothpick sequence of the third kind starting with a single toothpick.

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A194444 D-toothpick sequence of the second kind in the first quadrant.

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A160740 Toothpick sequence starting from a cross formed by 4 toothpicks.

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A194432 D-toothpick sequence starting with a cross formed by 4 toothpicks.

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A220514 D-toothpick sequence of the third kind starting with a X-shaped cross formed by 4 D-toothpicks.

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A212008 D-toothpick sequence of the second kind starting with a single toothpick.

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