A220514 -id:A220514 - cp's OEIS frontend

A221528 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220514.

0, 4, 8, 16, 16, 16, 32, 44, 32, 16, 32, 64, 96, 64, 80, 100, 64, 16, 32, 64, 96, 128, 144, 168, 176, 80, 96, 176, 256, 160, 176, 212, 128, 16, 32, 64, 96, 128, 144, 176, 208, 168, 192, 256, 400, 320, 336, 364, 336, 112, 96, 192, 288, 368, 416, 496

Offset: 0

Omar E. Pol, Mar 23 2013

Essentially the first differences of A220514.

First differs from A194435 at a(13).

			When written as an irregular triangle begins:
0;
4;
8;
16,16;
16,32,44,32;
16,32,64,96,64,80,100,64;
16,32,64,96,128,144,168,176,80,96,176,256,160,176,212,128;
16,32,64,96,128,144,176,208,168,192,256,400,320,336,364,336,112,96,192,288,368,416,496,528,216,224,400,576,352,368, 436,256;

Row lengths give 1 together with A011782.

Cf. A139251, A194435, A194701, A220501, A220521, A220523, A220525.

a(n) = 4*A220525(n).

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

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

A221528 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220514.

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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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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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