A160161 -id:A160161 - cp's OEIS frontend

A162798 a(n) = A160161(n+1)/2.

1, 2, 4, 4, 4, 4, 8, 16, 28, 16, 8, 4, 8, 16, 28, 28, 32, 40, 76, 116, 176, 72, 24, 16, 12, 20, 28, 28, 32, 40, 76, 116, 176, 108, 84, 88, 136, 180, 248, 224, 268, 332, 584, 744, 1000, 384, 152, 168, 132, 96, 56, 60, 64, 56, 84, 120, 176, 108, 84, 88, 136, 180, 248, 224

Offset: 1

Omar E. Pol, Jul 28 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, A139251, A160160, A160161, A162799.

A162799 a(n) = A160161(n+2)/4.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 8, 14, 8, 4, 2, 4, 8, 14, 14, 16, 20, 38, 58, 88, 36, 12, 8, 6, 10, 14, 14, 16, 20, 38, 58, 88, 54, 42, 44, 68, 90, 124, 112, 134, 166, 292, 372, 500, 192, 76, 84, 66, 48, 28, 30, 32, 28, 42, 60, 88, 54, 42, 44, 68, 90, 124, 112, 134, 166, 292, 372, 500, 246

Offset: 1

Omar E. Pol, Jul 28 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, A139251, A160160, A160161, A162798.

a(n) = A162798(n+1)/2.

A296612 Square array read by antidiagonals upwards: T(n,k) equals k times the number of compositions (ordered partitions) of n, with n >= 0 and k >= 1.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 8, 8, 6, 4, 5, 16, 16, 12, 8, 5, 6, 32, 32, 24, 16, 10, 6, 7, 64, 64, 48, 32, 20, 12, 7, 8, 128, 128, 96, 64, 40, 24, 14, 8, 9, 256, 256, 192, 128, 80, 48, 28, 16, 9, 10, 512, 512, 384, 256, 160, 96, 56, 32, 18, 10, 11, 1024, 1024, 768, 512, 320, 192, 112, 64, 36, 20, 11, 12

Offset: 0

Omar E. Pol, Jan 04 2018

Also, at least for the first five columns, column k gives the row lengths of the irregular triangles of the first differences of the total number of elements in the structure of some cellular automata. Indeed, the study of the structure and the behavior of the toothpick cellular automaton on triangular grid (A296510), and other C.A. of the same family, reveals that some cellular automata that have recurrent periods 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 an internal cycle. This internal cycle is called here "word" of a cellular automaton (see examples).

			The corner of the square array begins:
    1,   2,   3,    4,    5,    6,    7,    8,    9,   10, ...
    1,   2,   3,    4,    5,    6,    7,    8,    9,   10, ...
    2,   4,   6,    8,   10,   12,   14,   16,   18,   20, ...
    4,   8,  12,   16,   20,   24,   28,   32,   36,   40, ...
    8,  16,  24,   32,   40,   48,   56,   64,   72,   80, ...
   16,  32,  48,   64,   80,   96,  112,  128,  144,  160, ...
   32,  64,  96,  128,  160,  192,  224,  256,  288,  320, ...
   64, 128, 192,  256,  320,  384,  448,  512,  576,  640, ...
  128, 256, 384,  512,  640,  768,  896, 1024, 1152, 1280, ...
  256, 512, 768, 1024, 1280, 1536, 1792, 2048, 2304, 2560, ...
...
For k = 1 consider A160120, the Y-toothpick cellular automaton, which has word "a", so the structure of the irregular triangle of the first differences (A160161) is as follows:
a;
a;
a,a;
a,a,a,a;
a,a,a,a,a,a,a,a;
...
An associated sound to the animation of this cellular automaton could be (tick), (tick), (tick), ...
The row lengths of the above triangle are the terms of A011782, equaling the column 1 of the square array: 1, 1, 2, 4, 8, ...
.
For k = 2 consider A139250, the normal toothpick C.A. which has word "ab", so the structure of the irregular triangle of the first differences (A139251) 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;
...
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
The row lengths of the above triangle are the terms of A011782 multiplied by 2, equaling the column 2 of the square array: 2, 2, 4, 8, 16, ...
.
For k = 3 consider A296510, the toothpicks C.A. on triangular grid, which has word "abc", so the structure of the irregular triangle of the first differences (A296511) is as follows:
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;
...
An associated sound to the animation could be (tick, tock, tack), (tick, tock, tack), ...
The row lengths of the above triangle are the terms of A011782 multiplied by 3, equaling the column 3 of the square array: 3, 3, 6, 12, 24, ...
.
For k = 4 consider A299476, the toothpick C.A. on triangular grid with word "abcb", so the structure of the irregular triangle of the first differences (A299477) is as follows:
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;
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;
...
An associated sound to the animation could be (tick, tock, tack, tock), (tick, tock, tack, tock), ...
The row lengths of the above triangle are the terms of A011782 multiplied by 4, equaling the column 4 of the square array: 4, 4, 8, 16, 32, ...
.
For k = 5 consider A299478, the toothpick C.A. on triangular grid with word "abcbc", so the structure of the irregular triangle of the first differences (A299479) is as follows:
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;
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;
...
An associated sound to the animation could be (tick, tock, tack, tock, tack), (tick, tock, tack, tock, tack), ...
The row lengths of the above triangle are the terms of A011782 multiplied by 5, equaling the column 5 of the square array: 5, 5, 10, 20, 40, ...

Thomas Grubb and Frederick Rajasekaran, Set Partition Patterns and the Dimension Index, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.
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. A011782, A147562, A147582, A139250, A139251, A160160, A160161, A296510, A296511, A296610, A296611, A299476, A299477, A299478, A299479.

T(n,k) = k*A011782(n), with n >= 0 and k >= 1.

A160160 Toothpick sequence in the three-dimensional grid.

Original entry on oeis.org

0, 1, 3, 7, 15, 23, 31, 39, 55, 87, 143, 175, 191, 199, 215, 247, 303, 359, 423, 503, 655, 887, 1239, 1383, 1431, 1463, 1487, 1527, 1583, 1639, 1703, 1783, 1935, 2167, 2519, 2735, 2903, 3079, 3351, 3711, 4207, 4655, 5191, 5855, 7023, 8511, 10511, 11279, 11583, 11919, 12183, 12375, 12487, 12607

Offset: 0

Omar E. Pol, May 03 2009, May 06 2009

Similar to A139250, except the toothpicks are placed in three dimensions, not two. The first toothpick is in the z direction. Thereafter, new toothpicks are placed at free ends, as in A139250, perpendicular to the existing toothpick, but choosing in rotation the x-direction, y-direction, z-direction, x-direction, etc.

The graph of this sequence has a nice self-similar shape: it looks the when the x-range is multiplied by 2, e.g. a(0..125) vs a(0..250) or a(0..500). - M. F. Hasler, Dec 12 2018

M. F. Hasler, Table of n, a(n) for n = 0..500
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.]
R. J. Mathar, C++ program
R. J. Mathar, View after stage 1
R. J. Mathar, View after stage 2
R. J. Mathar, View after stage 3
R. J. Mathar, View after stage 4
R. J. Mathar, View after stage 5
R. J. Mathar, View after stage 6
R. J. Mathar, View after stage 7
R. J. Mathar, View after stage 8
R. J. Mathar, View after stage 9
R. J. Mathar, View after stage 10
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Alex van den Brandhof and Paul Levrie, Tandenstokerrij, Pythagoras, Viskundetijdschrift voor Jongeren, 55ste Jaargang, Nummer 6, Juni 2016, (see page 19 and the back cover).

Cf. A139250, A160120, A160161, A160170, A170884, A170885, A170876.

A160160_vec(n,o=1)={local(s(U)=[Vecsmall(Vec(V)+U)|V<-E], E=[Vecsmall([1,1,1])], J=[], M,A,B,U); [if(i>4, M+=8*#E=setminus(setunion(A=s(U=matid(3)[i%3+1,]), B=select(vecmin,s(-U))), J=setunion(setunion(setintersect(A,B),E),J)),M=1<M. F. Hasler, Dec 11 2018

A160160(n)=sum(k=1,n,A160161[k]) \\ if A160161=A160161_vec(n) has already been computed. - M. F. Hasler, Dec 12 2018

Partial sums of A160161: a(n) = Sum_{1 <= k <= n} A160161(k) for all n >= 0. - M. F. Hasler, Dec 12 2018

Edited by N. J. A. Sloane, Jan 02 2009
Extended to a(76) with C++ program and illustrations by R. J. Mathar, Jan 09 2010
Extended to 500 terms by M. F. Hasler, Dec 12 2018

A161645 First differences of A161644: number of new ON cells at generation n of the triangular cellular automaton described in A161644.

Original entry on oeis.org

0, 1, 3, 6, 6, 6, 12, 18, 12, 6, 12, 24, 30, 24, 30, 42, 24, 6, 12, 24, 30, 30, 42, 66, 66, 36, 30, 60, 84, 72, 78, 96, 48, 6, 12, 24, 30, 30, 42, 66, 66, 42, 42, 78, 114, 114, 114, 150, 138, 60, 30, 60, 84, 90, 114, 174, 198, 132, 90, 144, 210, 192, 192, 210, 96, 6, 12, 24

Offset: 0

David Applegate and N. J. A. Sloane, Jun 15 2009

See the comments in A161644.

It appears that a(n) is also the number of V-toothpicks or Y-toothpicks added at the n-th stage in a toothpick structure on hexagonal net, starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >=2 (see A161206, A160120, A182633). - Omar E. Pol, Dec 07 2010

			From _Omar E. Pol_, Apr 08 2015: (Start)
The positive terms written as an irregular triangle in which the row lengths are the terms of A011782:
1;
3;
6,6;
6,12,18,12;
6,12,24,30,24,30,42,24;
6,12,24,30,30,42,66,66,36,30,60,84,72,78,96,48;
6,12,24,30,30,42,66,66,42,42,78,114,114,114,150,138,60,30,60,84,90,114,174,198,132,90,144,210,192,192,210,96;
...
It appears that the right border gives A003945.
(End)

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]

Rémy Sigrist, Table of n, a(n) for n = 0..10000
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.]
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
N. J. A. Sloane, Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)
N. J. A. Sloane, Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version) [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A161644, A139251, A160161, A161207, A182633, A295559, A295560.

See A342271 for a(n)/3.

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
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See Links section.
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More terms from Rémy Sigrist, Jul 22 2022

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.

A160409 First differences of toothpick numbers A160408.

Original entry on oeis.org

1, 1, 2, 4, 4, 4, 4, 4, 8, 16, 16, 8, 4, 4, 8

Offset: 1

Omar E. Pol, May 23 2009

Number of toothpick added to the toothpick pyramid at the round n.

See also the toothpick sequences A139250, A160160 and the toothpick triangle A160406.

			Contribution from _Omar E. Pol_, Jun 06 2009: (Start)
Array begins:
========
x, y, z
========
1, 1, 2;
4, 4, 4;
4, 4, 8;
16, 16, 8;
4, 4, 8;
(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. A139250, A139251, A160160, A160161, A160406, A160407, A160408.

More terms from Omar E. Pol, Jun 06 2009

A160418 a(n) = A160407(n+2)/2.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 5, 4, 1, 1, 2, 2, 2, 3, 5, 4, 2, 3, 5, 5, 6, 10, 13, 8, 1, 1, 2, 2, 2, 3, 5, 4, 2, 3, 5, 5, 6, 10, 13, 8, 2, 3, 5, 5, 6, 10, 13, 9, 6, 10, 14, 15, 21, 32, 33, 16, 1, 1, 2, 2, 2, 3, 5, 4, 2, 3, 5, 5, 6, 10, 13, 8, 2, 3, 5, 5

Offset: 1

Omar E. Pol, May 23 2009

Row lengths are the terms of A000079 multiplied by 2. Right border gives A000079. - Omar E. Pol, Mar 19 2020

			From _Omar E. Pol_, Mar 19 2020: (Start)
Triangle begins:
  1,1;
  1,1,2,2;
  1,1,2,2,2,3,5,4;
  1,1,2,2,2,3,5,4,2,3,5,5,6,10,13,8;
  1,1,2,2,2,3,5,4,2,3,5,5,6,10,13,8,2,3,5,5,6,10,13,9,6,10,14,15,21,32,33,16;
  ... (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. A139250, A139251, A160160, A160161, A160406, A160407, A160408, A160409.

More terms from Jinyuan Wang, Mar 14 2020

A160729 First differences of A160728.

Original entry on oeis.org

6, 6, 12, 24, 24, 24, 24, 24, 48, 96, 96, 48, 24, 24, 48

Offset: 1

Omar E. Pol, Jul 28 2009

Also, 6 times A160409.

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, A160161, A160407, A160409, A160728.

a(n) = A160409(n)*6.

cp's OEIS Frontend

A162798 a(n) = A160161(n+1)/2.

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A162799 a(n) = A160161(n+2)/4.

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A296612 Square array read by antidiagonals upwards: T(n,k) equals k times the number of compositions (ordered partitions) of n, with n >= 0 and k >= 1.

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A160160 Toothpick sequence in the three-dimensional grid.

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PARI

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A161645 First differences of A161644: number of new ON cells at generation n of the triangular cellular automaton described in A161644.

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

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PARI

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A160171 First differences of X-toothpicks numbers A160170.

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A160409 First differences of toothpick numbers A160408.

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A160418 a(n) = A160407(n+2)/2.

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