A299477 -id:A299477 - cp's OEIS frontend

A139251 First differences of toothpicks numbers A139250.

0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 12, 12, 16, 28, 32, 16, 4, 8, 12, 12, 16, 28, 32, 20, 16, 28, 36, 40, 60, 88, 80, 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, 140, 208, 256, 192, 64, 4, 8, 12, 12, 16, 28, 32, 20, 16, 28

Offset: 0

Omar E. Pol, Apr 24 2008

Number of toothpicks added to the toothpick structure at the n-th step (see A139250).
It appears that if n is equal to 1 plus a power of 2 with positive exponent then a(n) = 4. (For proof see the second Applegate link.)
It appears that there is a relation between this sequence, even superperfect numbers, Mersenne primes and even perfect numbers. Conjecture: The sum of the toothpicks added to the toothpick structure between the stage A061652(k) and the stage A000668(k) is equal to the k-th even perfect number, for k >= 1. For example: A000396(1) = 2+4 = 6. A000396(2) = 4+4+8+12 = 28. A000396(3) = 16+4+8+12+12+16+28+32+20+16+28+36+40+60+88+80 = 496. - Omar E. Pol, May 04 2009
Concerning this conjecture, see David Applegate's comments on the conjectures in A153006. - N. J. A. Sloane, May 14 2009
In the triangle (See example lines), the sum of row k is equal to A006516(k), for k >= 1. - Omar E. Pol, May 15 2009
Equals (1, 2, 2, 2, ...) convolved with A160762: (1, 0, 2, -2, 2, 2, 2, -6, ...). - Gary W. Adamson, May 25 2009
Convolved with the Jacobsthal sequence A001045 = A160704: (1, 3, 9, 19, 41, ...). - Gary W. Adamson, May 24 2009
It appears that the sums of two successive terms of A160552 give the positive terms of this sequence. - Omar E. Pol, Feb 19 2015
From Omar E. Pol, Feb 28 2019: (Start)
The study of the toothpick 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 in general by irregular triangles (of first differences) whose row lengths are the terms of A011782 multiplied by k, where k >= 1, is the length of an internal cycle. This internal cycle is called "word" of a cellular automaton. For example: A160121 has word "a", so k = 1. This sequence has word "ab", so k = 2. A296511 has word "abc", so k = 3. A299477 has word "abcb" so k = 4. A299479 has word "abcbc", so k = 5.
The structure of this triangle (with word "ab" and k = 2) for the nonzero terms 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;
...
The row lengths are the terms of A011782 multiplied by 2, equaling the column 2 of the square array A296612: 2, 2, 4, 8, 16, ...
This arrangement has the property that the odd-indexed columns (a) contain numbers of the toothpicks that are parallel to initial toothpick, and the even-indexed columns (b) contain numbers of the toothpicks that are orthogonal to the initial toothpick (see the third triangle in the Example section).
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
For further information about the "word" of a cellular automaton see A296612. (End)

			From _Omar E. Pol_, Dec 16 2008: (Start)
Triangle begins:
1;
2;
4,4;
4,8,12,8;
4,8,12,12,16,28,32,16;
4,8,12,12,16,28,32,20,16,28,36,40,60,88,20,32;
(End)
From _David Applegate_, Apr 29 2009: (Start)
The layout of the triangle was adjusted to reveal that the columns become constant as shown below:
. 0;
. 1;
. 2,4;
. 4,4,8,12;
. 8,4,8,12,12,16,28,32;
.16,4,8,12,12,16,28,32,20,16,28,36,40,60,88,80;
.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,...
...
The row sums give A006516.
(End)
From _Omar E. Pol_, Feb 28 2018: (Start)
Also the nonzero terms can write as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 2 as shown below:
1,2;
4,4;
4,8,12,8;
4,8,12,12,16,28,32,16;
4,8,12,12,16,28,32,20,16,28,36,40,60,88,20,32;
...
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..65535
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.], 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.]
David Applegate, The movie version
Omar E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures) [From _Omar E. Pol_, Nov 02 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Equals 2*A152968 and 4*A152978 (if we ignore the first couple of terms).
See A147646 for the limiting behavior of the rows. See also A006516.
Row lengths in A011782.
Cf. A139250, A139252, A139253, A152980, A153000, A153001, A000396, A000668, A061652, A153006.
Cf. A006516, A153007, A159790, A001045, A160704, A160762, A147582, A296612.
Cf. A160121 (word "a"), A296511 (word "abc"), A299477 (word "abcb"), A299479 (word "abcbc").

G := (x/(1+2*x)) * (1 + 2*x*mul(1+x^(2^k-1)+2*x^(2^k),k=0..20)); # N. J. A. Sloane, May 20 2009, Jun 05 2009
# A139250 is T, A139251 is a.
a:=[0,1,2,4]; T:=[0,1,3,7]; M:=10;
for k from 1 to M do
a:=[op(a),2^(k+1)];
T:=[op(T),T[nops(T)]+a[nops(a)]];
for j from 1 to 2^(k+1)-1 do
a:=[op(a), 2*a[j+1]+a[j+2]];
T:=[op(T),T[nops(T)]+a[nops(a)]];
od: od: a; T;
# N. J. A. Sloane, Dec 25 2009

CoefficientList[Series[((x - x^2)/((1 - x) (1 + 2 x))) (1 + 2 x Product[1 + x^(2^k - 1) + 2 x^(2^k), {k, 0, 20}]), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 22 2014 *)

Recurrence from N. J. A. Sloane, Jul 20 2009: a(0) = 0; a(2^i)=2^i for all i; otherwise write n=2^i+j, 0 < j < 2^i, then a(n) = 2a(j)+a(j+1). Proof: This is a simplification of the following recurrence of David Applegate. QED
Recurrence from David Applegate, Apr 29 2009: (Start)
Write n=2^(i+1)+j, where 0 <= j < 2^(i+1). Then, for n > 3:
for j=0, a(n) = 2*a(n-2^i) (= n = 2^(i+1))
for 1 <= j <= 2^i - 1, a(n) = a(n-2^i)
for j=2^i, a(n) = a(n-2^i)+4 (= 2^(i+1)+4)
for 2^i+1 <= j <= 2^(i+1)-2, a(n) = 2*a(n-2^i) + a(n-2^i+1)
for j=2^(i+1)-1, a(n) = 2*a(n-2^i) + a(n-2^i+1)-4
and a(n) = 2^(n-1) for n=1,2,3. (End)
G.f.: (x/(1+2*x)) * (1 + 2*x*Product_{k>=0} (1 + x^(2^k-1) + 2*x^(2^k))). - N. J. A. Sloane, May 20 2009, Jun 05 2009
With offset 0 (which would be more natural, but offset 1 is now entrenched): a(0) = 1, a(1) = 2; for i >= 1, a(2^i) = 4; otherwise write n = 2^i +j, 0 < j < 2^i, then a(n) = 2 * Sum_{ k >= 0 } 2^(wt(j+k)-k)*binomial(wt(j+k),k). - N. J. A. Sloane, Jun 03 2009
It appears that a(n) = A187221(n+1)/2. - Omar E. Pol, Mar 08 2011
It appears that a(n) = A160552(n-1) + A160552(n), n >= 1. - Omar E. Pol, Feb 18 2015

Partially edited by Omar E. Pol, Feb 28 2019

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.

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.

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More terms from Rémy Sigrist, Jul 22 2022

A299476 Toothpick sequence on triangular grid with word "abcb".

Original entry on oeis.org

0, 1, 3, 7, 13, 21, 27, 39, 49, 57, 69, 87, 111, 135, 155, 185, 203, 211, 223, 241, 265, 293, 321, 367, 405, 453, 499, 567, 633, 701, 751, 823, 859, 869, 881, 899, 923, 951, 979, 1025, 1067, 1119, 1179, 1259, 1347, 1439, 1529, 1647, 1731, 1805

Offset: 0

Omar E. Pol, Feb 11 2018

a(n) gives the number of toothpicks of length 2 in the structure after n-th stage.
A299477 (the first differences) gives the number of toothpicks added at n-th stage.
For more information about the "word" of a cellular automaton see A296612.

Cf. A299477 (first differences), A296612.

Cf. A160120 (word "a"), A139250 (word "ab"), A296510 (word "abc"), A299478 (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").

cp's OEIS Frontend

A139251 First differences of toothpicks numbers A139250.

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

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A299476 Toothpick sequence on triangular grid with word "abcb".

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

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A355311 Number of V-toothpicks added at n-th stage to the V-toothpick structure of A355310.

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