A160731 -id:A160731 - cp's OEIS frontend

A160730 Toothpick sequence starting from a right angle formed by two toothpicks.

0, 2, 4, 8, 14, 20, 26, 38, 52, 64, 70, 82, 96, 112, 130, 162, 196, 216, 222, 234, 248, 264, 282, 314, 348, 372, 390, 422, 460, 504, 566, 658, 740, 776, 782, 794, 808, 824, 842, 874, 908, 932, 950, 982, 1020, 1064

Offset: 0

Omar E. Pol, May 25 2009

nonn

Nathaniel Johnston, Table of n, a(n) for n = 0..255
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
Nathaniel Johnston, C script

Cf. A139250, A160736, A160731.

Terms after a(11) from Nathaniel Johnston, Mar 28 2011

A323647 Number of elements added at n-th stage to the toothpick structure of A323646.

Original entry on oeis.org

1, 2, 2, 4, 6, 6, 6, 12, 14, 12, 6, 12, 14, 16, 18, 32, 34, 20, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 36, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 40, 18, 32, 38, 44, 62, 92, 86, 60, 62, 96, 114, 144, 210, 260, 194, 68, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92

Offset: 1

Omar E. Pol, Mar 07 2019

The "word" of this cellular automaton is "ab", but note that this triangle has an unusual structure: an additional row of length 2. For more information about the word of cellular automata see A296612.
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,a,b;
...
Row lengths give 2 together with the terms of A011782 multiplied by 2, also 2 togheter with the column 2 of A296612.
Columns "a" contain numbers of toothpicks of length 2.
Columns "b" contain numbers of D-toothpicks of length 2*sqrt(2). See the example.

			Triangle begins:
1, 2;
2, 4;
6, 6;
6,12,14,12;
6,12,14,16,18,32,34,20;
6,12,14,16,18,32,34,24,18,32,38,44,62,92,82,36;
6,12,14,16,18,32,34,24,18,32,38,44,62,92,82,40,18,32,38,44,62,92,86,60,62,96, ...

First differences of A323646.
Also, 1 together with A160731.
Column 1 gives A134201.
Cf. A011782, A139251, A296612.
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323651, A327331, A327333.

A160733 First differences of A160732.

Original entry on oeis.org

3, 3, 6, 8, 8, 6, 12, 14, 8, 8, 16, 20, 18, 18, 30, 26, 8, 8, 16, 20, 18, 20, 34, 32, 20, 28, 48, 54, 50, 60, 78, 50, 8, 8, 16, 20, 18, 20, 34, 32, 20, 28, 48, 54, 50, 62, 82, 56, 20, 28, 48, 54, 52, 70, 96

Offset: 1

Omar E. Pol, May 25 2009

nonn

Nathaniel Johnston, Table of n, a(n) for n = 1..274
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. A139250, A139251, A160731, A160732, A160737, A160739, A160741.

Terms after a(9) from Nathaniel Johnston, Mar 30 2011

A168113 First differences of A168112.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 6, 7, 6, 3, 6, 7, 8, 9, 16, 17, 10, 3, 6, 7, 8, 9, 16, 17, 12, 9, 16, 19, 22, 31, 46, 41, 18, 3, 6, 7, 8, 9, 16, 17, 12, 9, 16, 19, 22, 31, 46, 41, 20, 9, 16, 19, 22, 31, 46, 43, 30, 31, 48

Offset: 1

Omar E. Pol, Dec 07 2009

a(n) is the number of toothpicks added at the n-th round in the toothpick structure of A168112.

			If written as a triangle, begins:
1;
1;
2;
3,3;
3,6,7,6;
3,6,7,8,9,16,17,10;
3,6,7,8,9,16,17,12,9,16,19,22,31,46,41,18;
Rows converge to A168114.

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. A139250, A139251, A152978, A152980, A160407, A168112, A168114.

Observation: It appears that a(2^i+2) = 3, for i>0.

a(n) = A160731(n)/2. [From Nathaniel Johnston, Mar 28 2011]

Terms after a(34) from Nathaniel Johnston, Mar 28 2011

cp's OEIS Frontend

A160730 Toothpick sequence starting from a right angle formed by two toothpicks.

Views

Author

Keywords

Links

Crossrefs

Extensions

A323647 Number of elements added at n-th stage to the toothpick structure of A323646.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A160733 First differences of A160732.

Views

Author

Keywords

Links

Crossrefs

Extensions

A168113 First differences of A168112.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions