A168112 -id:A168112 - cp's OEIS frontend

A168113 First differences of A168112.

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

A323646 "Letter A" toothpick sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 21, 27, 39, 53, 65, 71, 83, 97, 113, 131, 163, 197, 217, 223, 235, 249, 265, 283, 315, 349, 373, 391, 423, 461, 505, 567, 659, 741, 777, 783, 795, 809, 825, 843, 875, 909, 933, 951, 983, 1021, 1065, 1127, 1219, 1301, 1341, 1359, 1391, 1429, 1473, 1535, 1627, 1713, 1773, 1835, 1931

Offset: 0

Omar E. Pol, Mar 07 2019

nonn

This arises from a hybrid cellular automaton formed of toothpicks of length 2 and D-toothpicks of length 2*sqrt(2).
For the construction of the sequence the rules are as follows:
On the infinite square grid at stage 0 there are no toothpicks, so a(0) = 0.
For the next n generations we have that:
At stage 1 we place a toothpick of length 2 in the horizontal direction, centered at [0,0], so a(1) = 1.
If n is even we add D-toothpicks. Each new D-toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.
If the x-coordinate of the middle point of the D-toothpick is negative then the D-toothpick must be placed in the NE-SW direction.
If the x-coordinate of the middle point of the D-toothpick is positive then the D-toothpick must be placed in the NW-SE direction.
If n is odd we add toothpicks in horizontal direction. Each new toothpick must have its midpoint touching the endpoint of exactly one existing D-toothpick.
The sequence gives the number of toothpicks and D-toothpicks after n stages.
A323647 (the first differences) gives the number of elements added at the n-th stage.
Note that if n >> 1 at the end of every cycle the structure looks like a "volcano", or in other words, the structure looks like a trapeze which is almost an isosceles right triangle.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

			After two generations the structure looks like a letter "A" which is formed by a initial I-toothpick (or a toothpick of length 2), placed in horizontal direction, and two D-toothpicks each of length 2*sqrt(2) as shown below, so a(2) = 3.
Note that angle between both D-toothpicks is 90 degrees.
.
                      *
                    *   *
                  * * * * *
                *           *
              *               *
.
After three generations the structure contains three horizontal toothpicks and two D-toothpicks as shown below, so a(3) = 5.
.
                      *
                    *   *
                  * * * * *
                *           *
          * * * * *       * * * * *
.

Cf. A139250, A168112, A160730, A296612, A323647.

For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A292612, A299770, A323650, A327330, A327332.

a(n) = 1 + A160730(n-1), n >= 1.

a(n) = 1 + 2*A168112(n-1), n >= 1.

A168114 If A168113 is regarded as a triangle then the rows converge to this sequence.

Original entry on oeis.org

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, 57, 72, 105, 130, 97, 36, 9, 16, 19, 22, 31, 46, 43, 30, 31, 48, 57, 72, 105, 130, 99, 46, 31, 48, 57, 72, 105, 132, 113, 88, 107, 150, 183, 246, 337, 354, 225

Offset: 1

Omar E. Pol, Dec 07 2009

nonn

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, A147646, A168112, A168113.

Terms after a(15) from Nathaniel Johnston, Mar 28 2011

cp's OEIS Frontend

A168113 First differences of A168112.

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A323646 "Letter A" toothpick sequence (see Comments for precise definition).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A168114 If A168113 is regarded as a triangle then the rows converge to this sequence.

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Author

Keywords

Links

Crossrefs

Extensions