A294980 -id:A294980 - cp's OEIS frontend

A294981 a(n) is the number of elements added at n-th stage to the structure of the cellular automaton of A294980.

1, 3, 6, 6, 6, 18, 18, 18, 6

Offset: 1

Omar E. Pol, Feb 10 2018

			This cellular automaton has word "ab". The row lengths are the terms of A011782 multiplied by 2, so the structure of this irregular triangle is as follows:
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;
...
Triangle begins:
1,  3;
6,  6;
6, 18, 18, 18;
...

Cf. A294980, A296612 (gives more information about the "word" of a cellular automaton).

Cf. A139251, A160121 (Y-toothpicks), A161207 (V-toothpicks).

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.

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.

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.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A294981 a(n) is the number of elements added at n-th stage to the structure of the cellular automaton of A294980.

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

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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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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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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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