A187211 -id:A187211 - cp's OEIS frontend

A188156 If A187211 is written, starting at its fifth term, as a triangle with rows of lengths 2, 4, 8, 16, ..., the n-th row begins with the first 2^n-1 terms of the present sequence.

22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 152, 70, 120, 150, 168, 246, 360, 342, 232, 246, 376, 454, 568, 838, 1032, 774, 280, 70, 120, 150, 168, 246, 360, 342, 232, 246, 376, 454, 568, 838, 1032

Offset: 1

Nathaniel Johnston, Mar 26 2011

nonn

Limiting behavior of the rows of the triangle in A187211.

			The triangle from A187211 begins:
22, 20
22, 40, 54, 40
22, 40, 54, 56, 70, 120, 134, 72
22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 136
...
Thus this sequence is 22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326...
The final entry of the n-th row (for n >= 2) is 16 + 8(2^n - 1).

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

Cf. A147646, A187210, A187211.

a(35) corrected by Nathaniel Johnston at the suggestion of Omar E. Pol, Mar 28 2011

A187210 Q-toothpick sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 5, 12, 24, 46, 66, 88, 128, 182, 222, 244, 284, 338, 394, 464, 584, 718, 790, 812, 852, 906, 962, 1032, 1152, 1286, 1374, 1444, 1564, 1714, 1882, 2128, 2488, 2814, 2950, 2972, 3012, 3066, 3122, 3192, 3312, 3446, 3534, 3604, 3724, 3874, 4042, 4288, 4648, 4974, 5126, 5196, 5316, 5466, 5634, 5880, 6240, 6582, 6814, 7060, 7436, 7890, 8458, 9296, 10328

Offset: 0

Omar E. Pol, Mar 07 2011

nonn

We define a "Q-toothpick" to be a quarter-circle. The length of a Q-toothpick is equal to Pi/2 = 1.570796...
In order to construct this sequence we use the following rules:
- Each new Q-toothpick must lie on the square grid (or circular grid) such that the Q-toothpick endpoints coincide with two opposite vertices of a unit square.
- Each exposed endpoint of the Q-toothpicks of the old generation must be touched by the endpoints of two q-toothpicks of new generation without creating a corner or vertex between these three arcs such that the couple of new Q-toothpicks should look like a "gullwing".
Note that in the Q-toothpick structure sometimes there is also an internal growth of the Q-toothpicks.
The sequence gives the number of Q-toothpicks in the structure after n stages. A187211 (the first differences) gives the number of Q-toothpicks added at n-th stage.
Note that the structure of the Q-toothpick cellular automaton contains distinct types of geometrical figures, for example: circles, diamonds, hearts, heads or flower vases (which appears only on the main diagonal) and also an infinity family of objects (blobs) where every object is a closed region which contains 2^k virtual circles with radius 1 and 2^k-1 virtual diamonds, for example: a 2 X 2 object is a closed region which contains exactly four virtual circles and three virtual diamonds, a 2 X 4 object is a closed region which contains exactly 8 virtual circles and 7 virtual diamonds, etc. Note that a "heart" can be considered a 1 X 2 object which contains two virtual circles and a virtual diamond. What is the better name for these figures? Note that there is a correspondence between this last family of objects and the squares and rectangles of the hidden crosses in the toothpick structure of A139250. For more information about the connection with the toothpick sequence see A139250, A160164 and A187220.
It appears that the number of hearts present in the n-th generation equals the number of rectangles of area = 2 present in the (n-2)nd generation of the toothpick structure of A139250, assuming the toothpicks have length 2, if n >= 3 (see also A188346 and A211008). - Omar E. Pol, Sep 30 2012
From Omar E. Pol, Jan 23 2016: (Start)
Consider the initial Q-toothpick with the virtual center at (0,0) and its endpoints at (0,1) and (1,0).
If n is a power of 2 plus 2 and n >> 1 then the structure of this C.A. essentially looks like a square which contains four parts (or sectors) as follows:
1) NW quadrant, but whose origin is at (-1,1). In this quadrant the number of Q-toothpicks after n generations equals the number of toothpicks in the toothpick structure of A139250 after n-2 generations, if n >= 2. Note that here the toothpick sequence A139250 is represented with Q-toothpicks arranged in an asymmetric structure.
2) SE quadrant, but whose origin is at (1,-1). This quadrant is a reflected copy of the NW quadrant, hence the number of Q-toothpicks after n generations equals A139250(n-2), n >= 2, the same as in the NW quadrant.
3) SW quadrant, but with the origin in the first quadrant at (1,1). In this quadrant the number of Q-toothpicks after n generations is 1 + A267694(n-1), n >= 1.
4) NE quasi-quadrant. In this sector the number of Q-toothpicks after n-generations is A267698(n-2) - 2, if n >= 6. (End)
After the first few generations the behavior is similar to the Gullwing cellular automaton of A187220, but the growth is faster than A187220 and thus it's much faster than A139250. For an animation see Applegate's The movie version in the Links section. - Omar E. Pol, Sep 13 2016

			From _Omar E. Pol_, Apr 02 2016: (Start)
Examples that are related to the toothpick sequence A139250 (see the first formula):
For n = 5 we have that A139250(5-2) = 7, A267698(5-2) = 13, A267694(5-1) = 16 and m = 3, so a(5) = 2*7 + 13 + 16 + 3 = 46.
For n = 6 we have that A139250(6-2) = 11, A267698(6-2) = 25, A267694(6-1) = 20 and m = -1, so a(6) = 2*11 + 25 + 20 - 1 = 66. (End)
From _Omar E. Pol_, Sep 13 2016: (Start)
Examples that are related to the Gullwing sequence A187220 (see the second formula):
For n = 5 we have that A187220(5-1) = 15, A267698(5-2) = 13, A267694(5-1) = 16 and m = 2, so a(5) = 15 + 13 + 16 + 2 = 46.
For n = 6 we have that A187220(6-1) = 23, A267698(6-2) = 25, A267694(6-1) = 20 and m = -2, so a(6) = 23 + 25 + 20 - 2 = 66. (End)

A. Adamatzky and G. J. Martinez, Designing Beauty: The Art of Cellular Automata, Springer, 2016, pages 59, 62 (note that the Q-toothpick cellular automaton is erroneously attributed to Nathaniel Johnston).

Nathaniel Johnston, Table of n, a(n) for n = 0..177
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.]
Elisabet Edvardsson and Eva Mossberg, The Q-toothpick Cellular Automaton, Journal of Cellular Automata, Vol. 14, Issue 1-2, (2019), p. 51-68.
Nathaniel Johnston, Animation of first 19 generations
Nathaniel Johnston, Illustration of a(5) = 46, "Front Matter" 2015. The College Mathematics Journal 46 (1). Mathematical Association of America: 1-1. doi:10.4169/college.math.j.46.1.fm.
Nathaniel Johnston, The Q-Toothpick Cellular Automaton
Nathaniel Johnston, The Q-Toothpick post in ConwayLife.com
Omar E. Pol, Illustration of initial terms
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

Cf. A139250, A160164, A187211, A187212, A187220, A188344, A188345, A188346, A267694, A267698.

a(0)=0; a(1)=1; a(n) = 2*A139250(n-2) + A267698(n-2) + A267694(n-1) + m, where m = 3 if 2 <= n <= 5 and m = -1 if n>=6 (note that 2*A139250(n-2) can be replaced with A160164(n-2)). - Omar E. Pol, Jan 23 2016

a(n) = A187220(n-1) + A267698(n-2) + A267694(n-1) + m, where m = 2 if 2 <= n <= 5 and m = -2 if n >= 6. - Omar E. Pol, Sep 13 2016

Terms a(8) and beyond from Nathaniel Johnston, Mar 26 2011
Comments edited by Omar E. Pol, Mar 28 2011
Second rule clarified by Omar E. Pol, Apr 06 2011

A187221 First differences of A187220.

Original entry on oeis.org

0, 1, 2, 4, 8, 8, 8, 16, 24, 16, 8, 16, 24, 24, 32, 56, 64, 32, 8, 16, 24, 24, 32, 56, 64, 40, 32, 56, 72, 80, 120, 176, 160, 64, 8, 16, 24, 24, 32, 56, 64, 40, 32, 56, 72, 80, 120, 176, 160, 72, 32, 56, 72, 80, 120, 176, 168, 112, 120, 184, 224, 280, 416, 512, 384, 128, 8

Offset: 0

Omar E. Pol, Mar 07 2011, Mar 09 2011

nonn

Number of gulls (or G-toothpicks) added at n-th stage to the gullwing structure of A187220.

Apparently this is the connection between A147582 and A139251. - Omar E. Pol, Mar 11 2011

			If written as an irregular triangle begins:
0,
1,
2,
4,
8,8,
8,16,24,16,
8,16,24,24,32,56,64,32,
8,16,24,24,32,56,64,40,32,56,72,80,120,176,160,64,
...
Also there is another version in which the layout of the irregular triangle was adjusted to reveal that the columns become constant:
.0,
.1,
.2,
.4,8,
.8,8,16,24,
16,8,16,24,24,32,56,64,
32,8,16,24,24,32,56,64,40,32,56,72,80,120,176,160,
64,8,16,24,24,32,56,64,40,32,56,72,80,120,176,160,72,32,56,72,80...

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

Cf. A139250, A139251, A147582, A187211, A187220.

a(0)=0. a(1)=1. It appears that a(n) = 2*A139251(n-1), for n >= 2.

A187213 Number of Q-toothpicks added at n-th stage to the structure of A187212.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 8, 10, 8, 4, 8, 12, 12, 16, 28, 30, 16, 4, 8, 12, 12, 16, 28, 32, 20, 16, 28, 36, 40, 60, 88, 78, 32, 4, 8, 12, 12, 16, 28, 32, 20, 16, 28, 36, 40, 60, 88, 80, 36, 16, 28, 36, 40, 60, 88, 84, 56, 60, 92, 112

Offset: 0

Omar E. Pol, Mar 22 2011

nonn

Essentially the first differences of A187212.

			Contribution from Omar E. Pol, Mar 29 2011 (Start):
If written as a triangle begins:
0,
1,
2,
2,4,
4,8,10,8,
4,8,12,12,16,28,30,16,
4,8,12,12,16,28,32,20,16,28,36,40,60,88,78,32,
(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.]
Nathaniel Johnston, The Q-Toothpick Cellular Automaton
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A187211, A187212, A187221.

It appears that if n = 2^k - 1, for k >= 2, then a(n) = A139251(n) - 2 otherwise a(n) = A139251(n). - Omar E. Pol, Mar 30 2011

Terms after a(24) from Nathaniel Johnston, Mar 28 2011

A267695 First differences of A267694.

Original entry on oeis.org

0, 2, 3, 4, 7, 4, 7, 12, 15, 4, 7, 12, 15, 12, 23, 36, 31, 4, 7, 12, 15, 12, 23, 36, 31, 12

Offset: 0

Omar E. Pol, Apr 02 2016

Number of Q-toothpicks added at n-th stage to the Q-toothpick structure of A267694.

			When the positive terms are written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
2;
3;
4, 7;
4, 7, 12, 15;
4, 7, 12, 15, 12, 23, 36, 31;
4, 7, 12, 15, 12, 23, 36, 31, 12,...

Cf. A011782, A187210, A187211, A267694, A267699.

A267699 First differences of A267698.

Original entry on oeis.org

0, 2, 4, 7, 12, 7, 12, 15, 20, 7, 12, 15, 20, 15, 28, 39, 36, 7, 12, 15, 20, 15, 28, 39, 36, 15

Offset: 0

Omar E. Pol, Apr 02 2016

Number of Q-toothpicks added at n-th stage in the Q-toothpick structure of A267698.

			When the positive terms are written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
2;
4;
7, 12;
7, 12, 15, 20;
7, 12, 15, 20, 15, 28, 39, 36;
7, 12, 15, 20, 15, 28, 39, 36, 15,...

Cf. A011782, A187210, A187211, A267695, A267698.

A187217 Number of Q-toothpicks added at n-th stage to the structure of A187216.

Original entry on oeis.org

0, 2, 6, 8, 14, 22, 30, 22, 38, 54, 70, 22, 38, 54, 70, 54, 102, 150, 134, 22, 38, 54, 70, 54, 102, 150, 134, 54, 102, 150, 166, 182, 326, 406, 262, 22, 38, 54, 70, 54, 102, 150, 134, 54, 102, 150, 166, 182, 326, 406, 262, 54, 102, 150, 166, 182, 326, 406, 294, 182, 326

Offset: 0

Omar E. Pol, Mar 30 2011

nonn

Essentially the first differences of A187216.

Nathaniel Johnston, Table of n, a(n) for n = 0..200
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.]
Nathaniel Johnston, The Q-Toothpick Cellular Automaton
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A187216, A187211, A187213, A187221.

a(15) - a(60) from Nathaniel Johnston, Apr 15 2011

A282470 Q-toothpick sequence with Q-toothpicks of radius 1 and 2 (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 16, 40, 62, 102, 124, 204, 258, 338, 360, 440, 494, 606, 676, 916, 1050, 1194, 1216, 1296, 1350, 1462, 1532, 1772, 1906, 2082, 2152, 2392, 2542, 2878, 3124, 3844, 4170, 4442, 4464, 4544, 4598, 4710, 4780, 5020, 5154, 5330, 5400, 5640, 5790, 6126, 6372, 7092, 7418, 7722, 7792, 8032, 8182, 8518

Offset: 0

Omar E. Pol, Feb 16 2017

nonn

For the construction of this sequence we use the same the rules of A187210 (the Q-toothpick sequence) except that for the even-indexed generations here we use Q-toothpicks of radius 2, not 1.
The result is that the structure looks like an arrangement of ovals.
On the infinite square grid at stage 0 we start with no Q-toothpicks, so a(0) = 0.
For n >= 1, a(n) is the ratio between the total length of the lines of the structure after n-th stages and the length of a single Q-toothpick of radius 1.
A187210(n) gives the total number of Q-toothpicks in the structure after n-th stages.
A187211(n) gives the number of Q-toothpicks added at n-th stage.
Note that since the radius of the Q-toothpicks can be two distincts numbers so we can write an infinite number of sequences from cellular automata of this kind.

Cf. A282471 (essentially the first differences).
Cf. A187210 (Q-toothpick sequence).
Cf. A139250, A187212, A267694, A267698.

A282471 First differences of A282470.

Original entry on oeis.org

0, 1, 8, 7, 24, 22, 40, 22, 80, 54, 80, 22, 80, 54, 112, 70, 240, 134, 144, 22, 80, 54, 112, 70, 240, 134, 176, 70, 240, 150, 336, 246, 720, 326, 272, 22, 80, 54, 112, 70, 240, 134, 176, 70, 240, 150, 336, 246, 720, 326, 304, 70, 240, 150, 336, 246, 720, 342, 464, 246, 752, 454, 1136, 838, 2064

Offset: 0

Omar E. Pol, Mar 17 2017

			Written as an irregular triangle the sequence begins:
0;
1;
8;
7;
24;
22, 40;
22, 80, 54, 80;
22, 80, 54, 112, 70, 240, 134, 144;
22, 80, 54, 112, 70, 240, 134, 176, 70, 240, 150, 336, 246, 720, 326, 272;
22, 80, 54, 112, 70, 240, 134, 176, 70, 240, 150, 336, 246, 720, 326, 304, 70,...
...
Starting from a(3) = 7 the row lengths of triangle are the terms of A011782.

Cf. A011782, A139251, A187210, A187211, A182470, A282470.

a(2n) = 2*A187211(2n).

a(2n+1) = A187211(2n+1).

A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 5, 4, 2, 4, 6, 6, 8, 14, 15, 8, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 39, 16, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 40, 18, 8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 95

Offset: 1

Omar E. Pol, Mar 22 2011, Apr 06 2011

nonn

It appears that both a(2) and a(2^k - 1) are odd numbers, for k >= 2. Other terms are even numbers.

			At stage 1 we start in the first quadrant from a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1). There are no gulls in the structure, so a(1) = 0.
At stage 2 we place a gull (or G-toothpick) with its midpoint at (1,1) and its endpoints at (2,0) and (2,2), so a(2) = 1. There is only one exposed midpoint at (2,2).
At stage 3 we place a gull with its midpoint at (2,2), so a(3) = 1. There are two exposed endpoints.
At stage 4 we place two gulls, so a(4) = 2. There are two exposed endpoints.
At stage 5 we place two gulls, so a(5) = 2. There are four exposed endpoints.
And so on.
If written as a triangle begins:
0,
1,
1,2,
2,4,5,4,
2,4,6,6,8,14,15,8,
2,4,6,6,8,14,16,10,8,14,18,20,30,44,39,16,
2,4,6,6,8,14,16,10,8,14,18,20,30,44,40,18,8,14,18,20,30,44,42,28,...
It appears that rows converge to A151688.

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. A099035, A151550, A151688, A187210, A187211, A187212, A187213, A187220.

read("transforms3") ;
L := BFILETOLIST("b187212.txt") ;
A187213 := DIFF(L) ;
seq( op(n,A187213)/2,n=2..nops(A187213)) ; # R. J. Mathar, Mar 30 2011

a(1)=0. a(n) = A187213(n)/2, for n >= 2.

It appears that a(2^k - 1) = A099035(k-1), for k >= 2.

cp's OEIS Frontend

A188156 If A187211 is written, starting at its fifth term, as a triangle with rows of lengths 2, 4, 8, 16, ..., the n-th row begins with the first 2^n-1 terms of the present sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A187210 Q-toothpick sequence (see Comments for precise definition).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A187221 First differences of A187220.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A187213 Number of Q-toothpicks added at n-th stage to the structure of A187212.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A267695 First differences of A267694.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A267699 First differences of A267698.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A187217 Number of Q-toothpicks added at n-th stage to the structure of A187216.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A282470 Q-toothpick sequence with Q-toothpicks of radius 1 and 2 (see Comments for precise definition).

Views

Author

Keywords

Comments

Links

Crossrefs

A282471 First differences of A282470.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.