A220524 -id:A220524 - cp's OEIS frontend

A220525 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220524.

0, 1, 2, 4, 4, 4, 8, 11, 8, 4, 8, 16, 24, 16, 20, 25, 16, 4, 8, 16, 24, 32, 36, 42, 44, 20, 24, 44, 64, 40, 44, 53, 32, 4, 8, 16, 24, 32, 36, 44, 52, 42, 48, 64, 100, 80, 84, 91, 84, 28, 24, 48, 72, 92, 104, 124, 132, 54, 56, 100, 144, 88, 92, 109, 64, 4

Offset: 0

Omar E. Pol, Mar 23 2013

Essentially the first differences of A220524.

First differs from A194445 at a(13).

			When written as an irregular triangle begins:
0;
1;
2;
4,4;
4,8,11,8;
4,8,16,24,16,20,25,16;
4,8,16,24,32,36,42,44,20,24,44,64,40,44,53,32;
4,8,16,24,32,36,44,52,42,48,64,100,80,84,91,84,28,24,48,72,92,104,124,132,54,56,100,144,88,92,109,64;

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

Cf. A139251, A194701, A194445, A220501, A220521, A220524, A220527.

a(n) = A221528(n)/4.

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

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

A231348 Number of triangles after n-th stage in a cellular automaton based in isosceles triangles of two sizes (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 33, 41, 45, 53, 65, 81, 91, 111, 133, 149, 153, 161, 173, 189, 201, 225, 253, 285, 295, 315, 343, 383, 405, 449, 495, 527, 531, 539, 551, 567, 579, 603, 631, 663, 675, 699, 731, 779, 807, 863, 923, 987, 997, 1017, 1045, 1085, 1113, 1169, 1233, 1313, 1335, 1379, 1439, 1527, 1573, 1665, 1759, 1823

Offset: 0

Omar E. Pol, Dec 15 2013

nonn

On the semi-infinite square grid the structure of this C.A. contains "black" triangles and "gray" triangles (see the Links section). Both types of triangles have two sides of length 5^(1/2). Every black triangle has a base of length 2 hence its height is 2 and its area is 2. Every gray triangle has a base of length 2^(1/2) hence its height is 3/(2^(1/2)) and its area is 3/2. Both types of triangles are arranged in the same way as the triangles of Sierpinski gasket (see A047999 and A006046). The black triangles are arranged in vertical direction. On the other hand the gray triangles are arranged in diagonal direction in the holes of the structure formed by the black triangles. Note that the vertices of all triangles coincide with the grid points.
The sequence gives the total number of triangles (black and gray) in the structure after n-th stage. A231349 (the first differences) gives the number of triangles added at n-th stage.
For a more complex structure see A233780.

			We start at stage 0 with no triangles, so a(0) = 0.
At stage 1 we add a black triangle, so a(1) = 1.
At stage 2 we add two black triangles, so a(2) = 1+2 = 3.
At stage 3 we add two black triangles and two gray triangles from the vertices of the master triangle, so a(3) = 3+2+2 = 7.
At stage 4 we add four black triangles, so a(4) = 7+4 = 11.
At stage 5 we add two black triangles and two gray triangles from the vertices of the master triangle, so a(5) = 11+2+2 = 15.
At stage 6 we add four black triangles and four gray triangles, so a(6) = 15+4+4 = 23.
At stage 7 we add four black triangles and six gray triangles, so a(7) = 23+4+6 = 33.
At stage 8 we add eight black triangles, so a(8) = 33+8 = 41.
And so on.
Note that always we add both black triangles and gray triangles except if n is a power of 2. In this case at stage 2^k we add only 2^k black triangles, for k >= 0.

Omar E. Pol, Illustration of the structure after 16 stages
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A047999, A006046, A139250, A151566, A160406, A173530, A182634, A194444, A220524, A220478, A230981, A231349, A233780.

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).

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.

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.

A231346 Number of distinct polygonal shapes after n-th stage in the structure of the D-toothpick cellular automaton of A220500.

Original entry on oeis.org

0, 0, 0, 1, 3, 4, 5, 7, 8, 8, 8, 11, 15, 17, 18, 19, 19, 19, 19, 19, 22

Offset: 1

Omar E. Pol, Dec 07 2013

The cellular automaton of A220500 contains a large number of distinct polygonal shapes. The exact number is unknown. Apparently it's greater than 63.
For simplicity we also call polygonal shapes "polygons".
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.
- Unfinished polygons with inward growth are not counted.
- If two polygons have the same shape but they have different size then these polygons must be counted as distinct polygonal shapes.
- The reflected shapes of asymmetric polygons, both with the same area, must be counted as distinct polygonal shapes.
Question: Is there a maximal record in this sequence?

Omar E. Pol, Illustration of the structure of A220500 after 17 stages, (Contains 19 distinct polygonal shapes.)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A194276, A220500, A220512, A220514, A220520, A220522, A220524, A220526.

cp's OEIS Frontend

A220525 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220524.

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

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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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A231348 Number of triangles after n-th stage in a cellular automaton based in isosceles triangles of two sizes (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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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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A220512 D-toothpick sequence of the third kind starting with a cross formed by 4 toothpicks.

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A231346 Number of distinct polygonal shapes after n-th stage in the structure of the D-toothpick cellular automaton of A220500.

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