A194444 -id:A194444 - cp's OEIS frontend

A194445 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194444.

0, 1, 2, 4, 4, 4, 8, 11, 8, 4, 8, 16, 24, 12, 20, 25, 16, 4, 8, 16, 24, 28, 36, 42, 44, 20, 24, 40, 64, 32, 44, 53, 32, 4, 8, 16, 24, 28, 36, 44, 52, 42, 48, 60, 100, 68, 84, 83, 84, 28, 24, 44, 72, 84, 104, 116, 132, 54, 56, 92, 144, 72, 92, 109, 64, 4

Offset: 0

Omar E. Pol, Aug 24 2011

Essentially the first differences of A194444.

First differs from A220525 at a(13). - Omar E. Pol, Mar 23 2013

			Contribution from _Omar E. Pol_, Dec 05 2012 (Start):
Triangle begins:
0;
1;
2;
4,4;
4,8,11,8;
4,8,16,24,12,20,25,16;
4,8,16,24,28,36,42,44,20,24,40,64,32,44,53,32;
(End)

Row lengths give 1 together with A011782. Right border gives 0 together with A000079.

Cf. A139250, A139251, A172311, A182635, A182839, A194271, A194433, A194435, A194441, A194443, A194444, A212009.

It appears that a(2^k+1) = 4, if k >= 1.

a(n) = A194435(n)/4. - Omar E. Pol, Mar 23 2013

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

A194440 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "wide" triangle of the second kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 27, 35, 39, 43, 51, 63, 79, 87, 103, 119, 123, 127, 135, 147, 163, 179, 203, 229, 253, 265, 285, 313, 353, 373, 405, 437, 441, 445, 453, 465, 481, 497, 521, 547, 571, 591, 623, 663, 727, 767, 815, 869, 909, 921, 941, 973, 1021

Offset: 0

Omar E. Pol, Aug 29 2011

nonn

For the D-toothpick "narrow" triangle of the second kind see A194442.

The structure is essentially one of the wedges of several D-toothpick structures. For more information see A194270. The first differences (A194441) give the number of toothpicks or D-toothpicks added at n-th stage. [Omar E. Pol, Dec 29 2012]

Omar E. Pol, Illustration of the structure after 32 stages
Omar E. Pol, Illustration of the structure after 64 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, A160120, A160406, A161830, A194270, A194277, A194441, A194442, A194444, A194692, A194693, A194696, A220494, A220520.

A194442 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the second kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 62, 70, 78, 91, 107, 111, 115, 123, 135, 151, 167, 187, 211, 223, 231, 247, 275, 291, 307, 332, 364, 368, 372, 380, 392, 408, 424, 446, 478, 504, 524, 548, 588, 620, 660, 693, 741, 761, 769, 785, 813, 853, 897, 947

Offset: 0

Omar E. Pol, Aug 29 2011

nonn

If n = 2^k, k >= 1, then the structure looks like an isosceles triangle. For the D-toothpick "wide" triangle of the second kind see A194440.

The structure is essentially one of the wedges of several D-toothpick structures. For more information see A194270. The first differences (A194443) give the number of toothpicks or D-toothpicks added at n-th stage. - Omar E. Pol, Mar 28 2013

Omar E. Pol, Illustration of the structure after 32 stages
Omar E. Pol, Illustration of the structure after 64 stages
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, A160120, A160406, A161830, A194270, A194277, A194440, A194443, A194444, A194694, A194695, A194697, A220496, A220522.

A220522 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the third kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 62, 70, 78, 91, 107, 111, 115, 123, 135, 151, 167, 187, 211, 223, 231, 247, 275, 291, 307, 332, 364, 368, 372, 380, 392, 408, 424, 446, 478, 504, 524, 548, 588, 620, 660, 701, 749, 769, 777

Offset: 0

Omar E. Pol, Dec 15 2012

nonn

The structure is essentially one of the oblique wedges of several D-toothpick structures. For more information see A220500. First differs from A194442 at a(47). The first differences (A220523) give the number of toothpicks or D-toothpicks added at n-th stage.

Omar E. Pol, Illustration of the structure after 32 stages
Omar E. Pol, Illustration of the structure after 64 stages
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, A194700, A194440, A194442, A194444, A220496, A220500, A220523.

A182838 H-toothpick sequence in the first quadrant starting with a D-toothpick placed on the diagonal [(0,1), (1,2)] (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 21, 31, 39, 43, 49, 61, 77, 91, 105, 127, 143, 147, 153, 165, 181, 197, 217, 249, 285, 307, 321, 349, 391, 431, 467, 517, 549, 553, 559, 571, 587, 603, 623, 655, 691, 715

Offset: 0

Omar E. Pol, Dec 12 2010

An H-toothpick sequence is a toothpick sequence on a square grid that resembles a partial honeycomb of hexagons.
The structure has two types of elements: the classic toothpicks with length 1 and the "D-toothpicks"  with length sqrt(2).
Classic toothpicks are placed in the vertical direction and D-toothpicks are placed in a diagonal direction.
Each hexagon has area = 4.
The network looks like an elongated hexagonal lattice placed on the square grid so that all nodes of the hexagonal net coincide with some of the grid points of the square grid. Each node in the hexagonal network is represented with coordinates x,y.
The sequence gives the number of toothpicks and D-toothpicks after n steps. A182839 (first differences) gives the number added at the n-th stage.
[It appears that for this sequence a classic toothpick is a line segment of length 1 that is parallel to the y-axis. A D-toothpick is a line segment of length sqrt(2) with slope +-1. D stands for diagonal. It also appears that classic toothpicks are not placed on the y-axis. - N. J. A. Sloane, Feb 06 2023]
From Omar E. Pol, Feb 17 2023: (Start)
This cellular automaton appears to be a version on the square grid of the first quadrant of the structure of A182840.
The rules are as follows:
- The elements (toothpicks and D-toothpicks) are connected at their ends.
- At each free end of the elements of the old generation two elements of the new generation must be connected.
- The toothpicks of length 1 must always be placed vertically, i.e. parallel to the Y-axis.
- The angle between a toothpick of length 1 and a D-toothpick of length sqrt(2) that share the same node must be 135 degrees, therefore the angle between two D-toothpicks that share the same node is 90 degrees.
As a result of these rules we can see that in the odd-indexed rows of the structure are placed only the toothpicks of length 1 and in the even-indexed rows of the structure are placed the D-toothpicks of length sqrt(2).
Apart from the trapezoids, pentagons and heptagons that are adjacent to the axes of the first quadrant it appears that there are only three types of polygons:
- Regular hexagons of area 4.
- Concave decagons (or concave 10-gons) of area 8.
- Concave dodecagons (or concave 12-gons) of area 12.
There are infinitely many of these polygons.
The structure shows a fractal-like behavior as we can see in other members of the family of toothpick cellular automata.
The structure has internal growth as some members of the mentioned family. (End)

			We start at stage 0 with no toothpicks.
At stage 1 we place a D-toothpick [(0,1),(1,2)], so a(1)=1.
At stage 2 we place a toothpick [(1,2),(1,3)] and a D-toothpick [(1,2),(2,1)], so a(2)=1+2=3.
At stage 3 we place 4 elements: a D-toothpick [(1,3),(0,4)], a D-toothpick [(1,3),(2,4)], a D-toothpick [(2,1),(3,2)] and a toothpick [(2,1),(2,0)], so a(3)=3+4=7. Etc.
The first hexagon appears in the structure after 4 stages.

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.]
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of a(32) = 549
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A153000, A161206, A170888, A172308, A182632, A182634, A182839, A182840, A187212, A194444, A220524, A233970, A267458, A267694, A267698.

See A360501 and A360512 for another hexagonal net built on the square grid. - N. J. A. Sloane, Feb 09 2023

Conjecture: a(n) = (A182840(n+1) + A267458(n+1) - 2)/4. - Omar E. Pol, Feb 10 2023

Partially edited by N. J. A. Sloane, Feb 06 2023

a(19)-a(41) from Omar E. Pol, Feb 06 2023

A194434 D-toothpick sequence of the second 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, 424, 504, 604, 668, 684, 716, 780, 876, 988, 1132, 1300, 1476, 1556, 1652, 1812, 2068, 2196, 2372, 2584, 2712, 2728, 2760, 2824, 2920, 3032, 3176, 3352, 3560, 3728, 3920, 4160, 4560, 4832, 5168

Offset: 0

Omar E. Pol, Sep 03 2011

nonn

On the infinite square grid we start with no toothpicks.
At stage 1, we place a cross as a "X", formed by 4 D-toothpicks of length sqrt(2) and centered at the origin. At stage 2, we place 8 toothpicks of length 1. At stage 3, we place 16 D-toothpicks of length sqrt(2). And so on.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194435) give the number of toothpicks or D-toothpicks added at n-th stage.
Apparently this cellular automaton has a fractal behavior (or fractal-like behavior) related to power of 2, similar to A194270 and very similar to A194432. The octagonal structure contains a large number of distinct closed polygonal regions. For more information see A194270, A194440 and A194442.
First differs from A220514 at a(13). - Omar E. Pol, Mar 23 2013
Observation: at least for the terms in the Data section the graph also appears to be a companion of the graph of A187210 but that could be different comparing more terms. - 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
Paolo Xausa, Animated version for n = 0..63
Paolo Xausa, Illustration of initial terms for n = 0..63 (multipage PDF)
Index entries for sequences related to toothpick sequences

Cf. A139250, A187210, A194270, A194432, A194435, A194440, A194442, A194444, A212008, A220514.

a(n) = 4*A194444(n).

More terms from Omar E. Pol, Mar 23 2013

A220524 D-toothpick sequence of the third kind in the first quadrant.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 34, 42, 46, 54, 70, 94, 110, 130, 155, 171, 175, 183, 199, 223, 255, 291, 333, 377, 397, 421, 465, 529, 569, 613, 666, 698, 702, 710, 726, 750, 782, 818, 862, 914, 956, 1004, 1068, 1168, 1248, 1332, 1423, 1507, 1535, 1559, 1607

Offset: 0

Omar E. Pol, Dec 15 2012

nonn

This is a toothpick sequence of forking paths to 135 degrees in the first quadrant. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A220525 (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 A194444 at a(13).

Cf. A139250, A194270, A194700, A194444, A220500, A220514, A220520, A220525.

a(n) = A220514(n)/4. - Omar E. Pol, Mar 23 2013

A194277 Known number of distinct polygonal shapes with n sides in the infinite D-toothpick structure of A194270.

Original entry on oeis.org

2, 4, 3, 6, 7, 2, 7, 7, 2, 3, 3, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1

Offset: 3

Omar E. Pol, Aug 25 2011

WARNING: The numbers are not fully tested. A new polygonal shape may appear in the structure beyond the stage 128 of A194270.
The cellular automaton of A194270 contains a large number of distinct polygonal shapes. For simplicity we call "polygons" to polygonal shapes.
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 (Note that some polygons contain in their body a toothpick or D-toothpick with an exposed endpoint; that element is not a part of the perimeter of the polygons).
- If two polygons have the same shape but they have different size then these polygons must be counted as distinct types of polygons.
- The reflected shapes of asymmetric polygons, both with the same area, must be counted as distinct types of polygons.
For more information see A194276 and A194278.

			Consider toothpicks of length 2 and D-toothpicks of length sqrt(2):
a(3) = 2 because the structure contains 2 types of triangles, each with area: 1, 2.
a(4) = 4 because the structure contains 4 types of quadrilaterals: 3 squares, each with area: 2, 4, 8 and also a rectangle with area 8.
a(5) = 3 because the structure contains 3 types of pentagons: a concave pentagon with area = 3 and also 2 convex pentagons with area 5 and 6.
a(12) = 3 because the structure contains 3 types of dodecagons: a symmetric concave dodecagon with area 29 and also 2 asymmetrict concave dodecagons both with area = 18. These last dodecagons are essentially equal but with reflected shape, so a(12) = 3 not 2.

Cf. A194270, A194276, A194278, A194444.

cp's OEIS Frontend

A194445 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194444.

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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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A194440 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "wide" triangle of the second kind.

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A194442 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the second kind.

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A220522 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the third kind.

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A182838 H-toothpick sequence in the first quadrant starting with a D-toothpick placed on the diagonal [(0,1), (1,2)] (see Comments for precise definition).

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A194434 D-toothpick sequence of the second kind starting with a X-shaped cross formed by 4 D-toothpicks.

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A220524 D-toothpick sequence of the third kind in the first quadrant.

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A194277 Known number of distinct polygonal shapes with n sides in the infinite D-toothpick structure of A194270.

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