A194700 -id:A194700 - cp's OEIS frontend

A194701 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194700.

0, 1, 4, 8, 16, 22, 24, 22, 40, 40, 32, 32, 56, 74, 96, 66

Offset: 0

Omar E. Pol, Sep 02 2011

nonn

Essentially the first differences of A194700. First differs from A194271 at a(15). Conjecture: this sequence and A194271 have infinitely many numbers in common.

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. A139251, A194271, A194700.

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

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.

A323650 Flower garden sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 15, 19, 27, 39, 63, 67, 75, 87, 111, 123, 147, 183, 255, 259, 267, 279, 303, 315, 339, 375, 447, 459, 483, 519, 591, 627, 699, 807, 1023, 1027, 1035, 1047, 1071, 1083, 1107, 1143, 1215, 1227, 1251, 1287, 1359, 1395, 1467, 1575, 1791, 1803, 1827, 1863, 1935, 1971, 2043, 2151, 2367, 2403, 2475

Offset: 0

Omar E. Pol, Jan 21 2019

nonn

This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks and V-toothpicks. Also, it appears that this is a missing link between A147562 (Ulam-Warburton) and three toothpick sequences: A139250 (normal toothpicks), A161206 (V-toothpicks) and A160120 (Y-toothpicks). The behavior resembles the toothpick sequence A139250, on the other hand, the formulas are directly related to A147562. Plot 2 shows that the graph is located between the graph of A139250 and the graph of A147562.
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 such that the angle of each single toothpick with respect to the connected I-toothpick is 120 degrees.
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.
A323651 (the first differences) gives the number of elements added at the n-th stage.
Note that 2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
The structure contains only three kinds of polygonal regions as follows:
- Rhombuses that contain two triangular cells.
- Regular hexagons that contain six triangular cells.
- Oblong hexagons that contain 10 triangular cells.
The structure looks like a "garden of flowers with six petals" (between other substructures). In particular, after 2^(n+1) stages with n >= 0, the structure looks like a flower garden in a rectangular box which contains A002450(n) flowers with six petals.
Note that this hybrid cellular automaton is also a superstructure of the Ulam-Warburton cellular automaton (at least in four ways). The explanation is as follows:
1) A147562(n) equals the total number of I-toothpicks in the structure after 2*n - 1 stage, n >= 1.
2) A147562(n) equals the total number of pairs of Y-toothpicks connected by their endpoints in the structure after 2*n stage (see the example).
3) A147562(n) equals the total number of "flowers with six petals" (or six-pointed stars formed of six rhombuses) in the structure after 4*n stage. Note that the location of the "flowers with six petals" in the structure is essentially the same as the location of the "ON" cells in the version "one-step bishop" of A147562.
4) For more connections to A147562 see the Formula section.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
The total number of “flowers with six petals” after n-th stage equals the total number of “hidden crosses” after n-th stage in the toothpick structure of A139250, including the central cross (beginning to count the crosses when their “nuclei” are totally formed with 4 quadrilaterals). - Omar E. Pol, Mar 06 2019

			Illustration of initial terms:
.
                        |   |
                \ /     |\ /|
         |       |        |
         |       |        |
                / \     |/ \|
                        |   |
n        1       2        3
a(n)     1       3        7
.
Note that for n = 2 the structure is also the same as a pair of Y-toothpicks connected by their endpoints (see A160120).

Cf. A002450, A103454, A139250 (normal toothpicks), A147562 (Ulam-Warburton), A147582, A160120 (Y-toothpicks), A161206 (V-toothpicks), A296612, A323641, A323642, A323649 (corner sequence), A323651 (first differences).

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

bw3[n_]:=bw3[n]=3^(DigitCount[n,2,1]-1);
A147562[n_]:=A147562[n]=If[n<2,n,1+4Sum[bw3[k],{k,n-1}]];
A323650[n_]:=If[OddQ[n],3A147562[(n-1)/2]+A147562[n]-A147562[n-1],3A147562[n/2]];
nterms=100;Array[A323650,nterms,0] (* Paolo Xausa, Jun 17 2022 *)

a(n) = 3*A147562(n/2) if n is even.
a(n) = 3*A147562((n-1)/2) + A147582(n) if n is odd.
a(n) = 3*A147562((n-1)/2) + A147562(n) - A147562(n-1) if n is odd.
a(2^n) = A103454(n), n >= 0.

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

A294962 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton formed by toothpicks, D-toothpicks and T-toothpicks on the infinite square grid (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 5, 13, 21, 33, 53, 69, 77, 101, 117

Offset: 0

Omar E. Pol, Nov 12 2017

It seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.
After 10th stage there are no exposed endpoints in the structure, so the structure is finished.
A294963(n) gives the number of elements added to the structure at n-th stage.

Cf. A139250 (toothpicks), A160172 (T-toothpicks), A194700 (D-toothpicks), A220500, A294963.

For other hybrid cellular automata, see A289840, A290220, A294020, A294980.

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).

A299770 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton on the infinite square grid, formed by toothpicks of length 2, D-toothpicks, toothpicks of length 1, and T-toothpicks.

Original entry on oeis.org

1, 5, 13, 21, 33, 49, 65, 73, 97, 105

Offset: 1

Omar E. Pol, Mar 20 2018

The structure is essentially the same as the finite structure described in A294962 but here there are no D-toothpicks of length sqrt(2)/2. All D-toothpicks in the structure have length sqrt(2).
The same as A294962, it seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.
After 10th stage there are no exposed endpoints (or free ends), so the structure is finished.
A299771(n) gives the number of elements added to the structure at n-th stage.
The "word" of this cellular automaton is "abcd". For further information about the word of cellular automata see A296612. - Omar E. Pol, Mar 05 2019

Very similar to A294962.
Cf. A299771, A296612.
Cf. A139250 (toothpicks), A160172 (T-toothpicks), A194700 (D-toothpicks), A220500.
For other hybrid cellular automata, see A194270, A220500, A289840, A290220, A294020, A294980.

A220496 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "narrow" triangle of the first kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 58, 66, 74, 87, 103, 107, 111, 119, 127, 135, 143, 157, 173, 181, 189, 205, 221, 237, 253, 278, 310, 314, 318, 326, 334, 342, 350, 364, 380, 388, 396, 412, 428, 444, 460, 486, 518, 526, 534, 550, 566, 582

Offset: 0

Omar E. Pol, Dec 23 2012

nonn

This cellular automaton uses 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, in the first quadrant, we start with no elements, so a(0) = 0. At stage 1, we place a D-toothpick at (0,0),(1,1), so a(1) = 1. The rules for adding new elements are 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. The endpoints of the D-toothpicks of the old generation that are perpendiculars to the initial D-toothpick remain exposed forever. Overlapping is prohibited.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A220497 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage.
It appears that the structure has fractal behavior related to powers of 2. It appears that this cellular automaton has a surprising connection with the Sierpinski triangle, but here the structure is more complex.
For a similar version see A220494. For other more complex versions see A194442, A220522.
First differs from A194442 (and from A220522) at a(12).

Omar E. Pol, Illustration of the structure after 32 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, A194442, A194700, A220494, A220497, A220522.

A220526 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "medium" triangle of the third kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 62, 76, 88, 103, 119, 123, 127, 135, 147, 163, 183, 207, 233

Offset: 0

Omar E. Pol, Jan 02 2013

The structure is essentially one of the horizontal wedges of A220500. First differs from A194442 (and from A220522) at a(13). A220527 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage.

Cf. A139250, A194270, A194440, A194700, A194442, A220500, A220512, A220514, A220520, A220522, A220524, A220527.

cp's OEIS Frontend

A194701 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194700.

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

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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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A323650 Flower garden sequence (see Comments for precise definition).

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

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A294962 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton formed by toothpicks, D-toothpicks and T-toothpicks on the infinite square grid (see Comments lines for precise definition).

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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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A299770 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton on the infinite square grid, formed by toothpicks of length 2, D-toothpicks, toothpicks of length 1, and T-toothpicks.

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A220496 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "narrow" triangle of the first kind.

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A220526 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "medium" triangle of the third kind.

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