A221528 -id:A221528 - cp's OEIS frontend

A220514 D-toothpick sequence of the third kind starting with a X-shaped cross formed by 4 D-toothpicks.

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).

A194435 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194434.

Original entry on oeis.org

0, 4, 8, 16, 16, 16, 32, 44, 32, 16, 32, 64, 96, 48, 80, 100, 64, 16, 32, 64, 96, 112, 144, 168, 176, 80, 96, 160, 256, 128, 176, 212, 128, 16, 32, 64, 96, 112, 144, 176, 208, 168, 192, 240, 400, 272, 336, 332, 336, 112, 96, 176, 288, 336, 416, 464

Offset: 0

Omar E. Pol, Sep 03 2011

Essentially the first differences of A194434.
First differs from A221528 at a(13). - Omar E. Pol, Mar 23 2013
From Omar E. Pol, Jun 24 2022: (Start)
The word of this cellular automaton is "ab".
For the nonzero terms the structure of the irregular triangle is as shown below:
  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;
  ...
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
Columns "a" contain numbers of D-toothpicks (of length sqrt(2)).
Columns "b" contain numbers of toothpicks (of length 1).
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
For further information about the word of cellular automata see A296612. (End)

			From _Omar E. Pol_, Mar 23 2013: (Start)
When written as an irregular triangle the sequence of nonzeros terms begins:
   4, 8;
  16,16;
  16,32,44,32;
  16,32,64,96, 48, 80,100, 64;
  16,32,64,96,112,144,168,176, 80, 96,160,256,128,176,212,128;
  16,32,64,96,112,144,176,208,168,192,240,400,272,336,332,336,112,96, ...
  ... (End)
Right border gives the powers of 2 >= 8 (reformatted the triangle). - _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 (red elements)
Paolo Xausa, Animated version for n = 0..63 (red elements)
Paolo Xausa, Illustration of initial terms for n = 0..63 (red elements, multipage PDF)
Index entries for sequences related to toothpick sequences

Cf. A011782, A139251, A194271, A194433, A194434, A194441, A194443, A194445, A212009, A221528, A299612.

a(n) = 4*A194445(n).

Conjecture: a(2^k+1) = 16, if k >= 1.

More terms from Omar E. Pol, Mar 23 2013

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

Original entry on oeis.org

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.

A221565 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220512.

Original entry on oeis.org

0, 4, 8, 16, 16, 16, 28, 48, 32, 16, 32, 64, 64, 80, 84, 112, 64, 16, 32, 64, 96, 112, 144, 176, 136, 112, 96, 176, 176, 208, 196, 240, 128, 16, 32, 64, 96, 112, 152

Offset: 0

Omar E. Pol, May 13 2013

The first differences of A220512.

First differs from A194433 at a(14).

			When written as a irregular triangle begins:
0;
4;
8;
16,16;
16,28,48,32;
16,32,64,64,80,84,112,64;
16,32,64,96,112,144,176,136,112,96,176,176,208,196,240,128;
16,32,64,96,112,152...

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

Cf. A139251, A194271, A194701, A194443, A220501, A220512, A220521, A220523, A220525, A220527, A221528.

cp's OEIS Frontend

A220514 D-toothpick sequence of the third kind starting with a X-shaped cross formed by 4 D-toothpicks.

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A194435 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194434.

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

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A221565 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220512.

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