A160121 -id:A160121 - cp's OEIS frontend

A182841 Number of toothpicks added at n-th stage in the toothpick structure of A182840.

0, 1, 4, 8, 14, 16, 14, 24, 38, 32, 14, 24, 46, 64, 54, 56, 86, 64, 14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128, 14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256

Offset: 0

Omar E. Pol, Dec 09 2010

First differences of A182840.

			From _Omar E. Pol_, Nov 01 2014: (Start)
When written as an irregular triangle:
0;
1;
4;
8;
14, 16;
14, 24, 38, 32;
14, 24, 46, 64, 54, 56, 86, 64;
14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128;
14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256;
...
(End)

Olaf Voß, Table of n, a(n) for n = 0..1000
David Applegate, The movie version
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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Olaf Voß, Toothpick structures on hexagonal net
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A139251, A160121, A161207, A182633, A182635, A151724, A182842.

a(2^k + 1) = 2^(k+2), at least for 0 <= k <= 9. - Omar E. Pol, Nov 01 2014

More terms from Olaf Voß, Dec 24 2010

Wiki link added by Olaf Voß, Jan 14 2011

A182635 Number of toothpicks added at n-th stage to the toothpick structure of A182634.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 16, 20, 16, 20, 28, 16, 4, 8, 16, 20, 20, 28, 44, 44, 24, 20, 40, 56, 48, 52, 64, 32, 4, 8, 16, 20, 20, 28, 44, 44, 28, 28, 52, 76, 76, 76

Offset: 0

Omar E. Pol, Dec 08 2010

First differences of A182634.

First differs from A139251 at a(11).

			Contribution from _Omar E. Pol_, Dec 06 2012 (Start):
When written as an irregular triangle begins:
0;
1;
2;
4,4;
4,8,12,8;
4,8,16,20,16,20,28,16;
4,8,16,20,20,28,44,44,24,20,40,56,48,52,64,32;
4,8,16,20,20,28,44,44,28,28,52,76,76,76,...
(End)

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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A139251, A160121, A161207, A161645, A182633, A182634, A194445.

a(n) = A182633(n)/3.

A323651 Number of elements added at n-th stage to the toothpick structure of A323650.

Original entry on oeis.org

1, 2, 4, 8, 4, 8, 12, 24, 4, 8, 12, 24, 12, 24, 36, 72, 4, 8, 12, 24, 12, 24, 36, 72, 12, 24, 36, 72, 36, 72, 108, 216, 4, 8, 12, 24, 12, 24, 36, 72, 12, 24, 36, 72, 36, 72, 108, 216, 12, 24, 36, 72, 36, 72, 108, 216, 36, 72, 108, 216, 108, 216, 324, 648, 4, 8, 12, 24, 12, 24, 36, 72, 12, 24, 36, 72, 36, 72, 108, 216

Offset: 1

Omar E. Pol, Feb 04 2019

nonn

The odd-indexed terms (a bisection) gives A147582, the first differences of A147562 (Ulam-Warburton cellular automaton).
The even-indexed terms (a bisection) gives A147582 multiplied by 2.
The word of this cellular automaton is "ab", so 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 I-toothpicks. Columns "b" contain numbers of V-toothpicks. See the example.
For further information about the word of cellular automata see A296612.

			Written as an irregular triangle the sequence begins:
1,2;
4,8;
4,8,12,24;
4,8,12,24,12,24,36,72;
4,8,12,24,12,24,36,72,12,24,36,72,36,72,108,216;
4,8,12,24,12,24,36,72,12,24,36,72,36,72,108,216,12,24,36,72,36,72,108,216,...
...

First differences of A323650.
Cf. A011782, A139250, A139251, A147562, A147582, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649.
For other hybrid cellular automata, see A194701, A194271, A220501, A290221, A294021, A294981.

a(2n-1) = A147582(n).
a(2n) = 2*A147582(n).
a(n) = 4*A323641(n-2), n >= 3.

A151710 First differences of A160715.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 21, 3, 9, 9, 21, 15, 21, 27, 51, 3, 9, 9, 21, 15, 21, 27, 51, 21, 21, 27, 51

Offset: 1

Omar E. Pol, Jun 02 2009

Number of Y-toothpick added at n-th stage to the structure of A160715.

Similar to A160121. [From Omar E. Pol, May 29 2010]

			Contribution from _Omar E. Pol_, Dec 18 2012 (Start):
Written as an irregular triangle begins:
1;
3;
3, 9;
3, 9, 9, 21;
3, 9, 9, 21, 15, 21, 27, 51;
3, 9, 9, 21, 15, 21, 27, 51, 21, 21, 27, 51,...
(End)

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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A000225, A006516, A139250, A160120.

Cf. A048883, A139251, A151837, A160121.

More terms from Omar E. Pol, May 29 2010

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

Original entry on oeis.org

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).

A296511 Number of toothpicks added at n-th stage to the toothpick structure of A296510.

Original entry on oeis.org

1, 2, 4, 6, 6, 6, 6, 10, 16, 20, 16, 10, 6, 10, 16, 24, 28, 32, 28, 32, 40, 50, 40, 22, 8, 10, 16, 24, 28, 32, 32, 40, 56, 74, 76, 64, 42, 36, 40, 62, 76, 90, 80, 88, 102, 122, 96, 50, 14, 10, 16, 24, 28, 32, 32, 40, 56, 74, 76, 64, 46, 44, 56, 82, 104, 124

Offset: 1

Omar E. Pol, Dec 14 2017

The structure and the behavior of this cellular automaton reveals that some cellular automata have recurrent periods that can be represented by irregular triangles of first differences whose row lengths are the terms of A011782 multiplied by k (instead of powers of 2), where k is the length of their "word". In this case the word must be "abc", therefore k = 3. In the case of the cellular automaton with normal toothpicks (A139250) the word must be "ab" and k = 2.
The associated sound to the animation of this cellular automaton could be [tick, tock, tack], [tic, tock, tack], and so on.
For more information about the "word" of a cellular automaton see A296612.

			The structure of this irregular triangle is as shown below:
   a, b, c;
   a, b, c;
   a, b, c, a, b, c;
   a, b, c, a, b, c, a, b, c, a, b, c;
   a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c;
...
Every column is associated successively to one of the axes of the triangular grid.
Every row represents a geometric period of the cellular automaton.
So, written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 3, the sequence begins:
   1, 2, 4;
   6, 6, 6;
   6,10,16,20,16,10;
   6,10,16,24,28,32,28,32,40,50,40,22;
   8,10,16,24,28,32,32,40,56,74,76,64,42,36,40,62,76,90,80,88,102,122,96,50;
  14,10,16,24,28,32,32,40,56,74,76,64,...
...

Rémy Sigrist, Table of n, a(n) for n = 1..6144
Rémy Sigrist, Illustration of the construction at generation 3*256
Rémy Sigrist, PARI program
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

First differences of A296510.
Cf. A160121 (word "a"), A139251 (word "ab"), A299477 (word "abcb"), A299479 (word "abcbc").
Cf. A007283, A011782, A151906, A160121, A160161, A292612.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Jul 22 2022

A299477 Number of toothpicks added at n-th stage to the structure of the cellular automaton of A299476.

Original entry on oeis.org

1, 2, 4, 6, 8, 6, 12, 10, 8, 12, 18, 24, 24, 20, 30, 18, 8, 12, 18, 24, 28, 28, 46, 38, 48, 46, 68, 66, 68, 50, 72, 36, 10, 12, 18, 24, 28, 28, 46, 42, 52, 60, 80, 88, 92, 90, 118, 84, 74

Offset: 1

Omar E. Pol, Feb 11 2018

The "word" of this cellular automaton is abcb.
The associated sound to the animation of this cellular automaton could be [tick, tock, tack, tock], [tick, tock, tack, tock], and so on.
For more information about the "word" of a cellular automaton see A296612.

			The structure of this irregular triangle is shown below:
   a,  b,  c,  b;
   a,  b,  c,  b;
   a,  b,  c,  b,  a,  b,  c,  b;
   a,  b,  c,  b,  a,  b,  c,  b,  a,  b,  c,  b,  a,  b,  c,  b;
...
So, written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 4, the sequence begins:
   1,  2,  4,  6;
   8,  6, 12, 10;
   8, 12, 18, 24, 24, 20, 30, 18;
   8, 12, 18, 24, 28, 28, 46, 38, 48, 46, 68, 66, 68, 50, 72, 36;
  10, 12, 18, 24, 28, 28, 46, 42, 52, 60, 80, 88,92, 90, 118, 84, 74, ...

Cf. A139251, A160121, A296511 (word "abc"), A299476, A299479 (word "abcbc").

A299479 Number of toothpicks added at n-th stage to the structure of the cellular automaton of A299478.

Original entry on oeis.org

1, 2, 4, 6, 8, 6, 6, 10, 16, 20, 12, 10, 16, 20, 32, 24, 22, 34, 36, 40, 16, 14, 20, 24, 40, 32, 32, 44, 48, 64, 48, 42, 58, 70, 104, 74, 60, 78, 78, 80, 24, 14, 20, 24, 40, 36, 40, 50, 52, 72

Offset: 1

Omar E. Pol, Feb 11 2018

The "word" of this cellular automaton is abcbc.
The associated sound to the animation of this cellular automaton could be [tick, tock, tack, tock, tack], [tick, tock, tack, tock, tack], and so on.
For more information about the "word" of a cellular automaton see A296612.

			The structure of this irregular triangle is shown below:
   a,  b,  c,  b,  c;
   a,  b,  c,  b,  c;
   a,  b,  c,  b,  c,  a,  b,  c,  b,  c;
   a,  b,  c,  b,  c,  a,  b,  c,  b,  c,  a,  b,  c,  b,  c,   a,  b,  c,  b,  c;
...
So, written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 5, the sequence begins:
   1,  2,  4,  6,  8;
   6,  6, 10, 16, 20;
  12, 10, 16, 20, 32, 24, 22, 34, 36, 40;
  16, 14, 20, 24, 40, 32, 32, 44, 48, 64, 48, 42, 58, 70, 104, 74, 60, 78, 78, 80;
  24, 14, 20, 24, 40, 36, 40, 50, 52, 72, ...

Cf. A299478.

Cf. A160121 (word "a"), A139251 (word "ab"), A296511 (word "abc"), A299477 (word "abcb").

A160171 First differences of X-toothpicks numbers A160170.

Original entry on oeis.org

0, 1, 4, 8, 8, 24, 32, 32, 56, 80, 80, 88, 112, 160, 168, 240, 224, 344, 320, 320, 344, 448, 528, 536, 704, 704, 872, 736, 880, 840, 1024, 1088, 1256, 1328, 1392, 1416, 1440, 1504, 1656

Offset: 0

Omar E. Pol, May 03 2009, Dec 13 2010

nonn

Number of X-toothpicks added at n-th stage to the three-dimensional X-toothpick structure of A160170.

For another version see A170875.

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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

X-toothpick sequence: A160170.

Cf. A139250, A139251, A160120, A160121, A160160, A160161, A170875.

More terms (a(6)-a(38)) based on Email from R. J. Mathar dated on Jan 10 2010.

A173452 a(n) = A048883(n-1) - A151710(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, -6, 6, 0, 30, 0, 0, 0, 6, -6, 6, 0, 30, -12, 6, 0, 30

Offset: 1

Omar E. Pol, May 29 2010

It appears that the absolute value of a(n) is a multiple of 6, see A008588. - Omar E. Pol, Dec 06 2013

			From _Omar E. Pol_, Dec 06 2013: (Start)
Written as an irregular triangle in which row lengths is A011782 the sequence begins:
0;
0;
0, 0;
0, 0, 0, 6;
0, 0, 0, 6, -6, 6, 0, 30;
0, 0, 0, 6, -6, 6, 0, 30, -12, 6, 0, 30...
(End)

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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A048883, A139250, A139251, A160121, A151710, A173066, A173067, A173068, A173451, A173453.

cp's OEIS Frontend

A182841 Number of toothpicks added at n-th stage in the toothpick structure of A182840.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A182635 Number of toothpicks added at n-th stage to the toothpick structure of A182634.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A323651 Number of elements added at n-th stage to the toothpick structure of A323650.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A151710 First differences of A160715.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

A296511 Number of toothpicks added at n-th stage to the toothpick structure of A296510.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A299477 Number of toothpicks added at n-th stage to the structure of the cellular automaton of A299476.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A299479 Number of toothpicks added at n-th stage to the structure of the cellular automaton of A299478.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A160171 First differences of X-toothpicks numbers A160170.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions