A160121 -id:A160121 - cp's OEIS frontend

A160123 a(n) = A160121(n+1)/3.

1, 1, 3, 1, 3, 3, 7, 3, 3, 3, 7, 5, 7, 9, 17, 9, 3, 3, 7, 5, 7, 9, 17, 11, 7, 9, 17, 17, 19, 23, 39, 27, 7, 3, 7, 5, 7, 9, 17, 11, 7, 9, 17, 17, 19, 23, 39, 29, 11, 9, 17, 17, 19, 25, 43, 39, 25, 23, 39, 45, 47, 57, 93, 77, 23, 3, 7, 5, 7, 9

Offset: 1

Omar E. Pol, May 02 2009, May 21 2010

nonn

Also, first differences of A161910. [From Omar E. Pol, Jun 21 2009]

			Note that this can be written as a triangle:
1;
1;
3;
1,3;
3,7,3,3;
3,7,5,7,9,17,9,3;
3,7,5,7,9,17,11,7,9,17,17,19,23,39,27,7;
3,7,5,7,9,17,11,7,9,17,17,19,23,39,29,11,9,17,17,19,25,43,39,25,23,39,45,47,57,93,77,23;
3,7,5,7,9,...

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

Y-toothpick sequence: A160120. Cf. A160121, A160122.
Cf. A139250, A139251. [From Omar E. Pol, Jun 16 2009]
Cf. A161831, A161427, A161910. [From Omar E. Pol, Jun 21 2009]

More terms from Omar E. Pol, Jun 16 2009

Extended by R. J. Mathar, Feb 05 2010

A173451 a(n) = A048883(n-1) - A160121(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 6, -6, 6, 0, 30, -24, 0, 0, 6, -6, 6, 0, 30, -24, 6, 0, 30, -24, 24, 12, 126, -78, -12, 0, 6, -6, 6, 0, 30, -24, 6, 0, 30, -24, 24, 12, 126, -78, -6, 0, 30, -24, 24, 6, 114, -90, 6, 12, 126, -54, 102, 72, 450, -228, -60, 0, 6, -6, 6, 0

Offset: 1

Omar E. Pol, May 29 2010

It appears that the absolute value of a(n) is a multiple of 6, see A008588. - Omar E. Pol, Dec 06 2013

			From _Omar E. Pol_, Dec 06 2013: (Start)
Written as an irregular triangle in which row lengths is A011782 the sequence begins:
.    0;
.    0;
.    0,  0;
.    0,  0,0,6;
.   -6,  0,0,6,-6,6,0,30;
.  -24,  0,0,6,-6,6,0,30,-24,6,0,30,-24,24,12,126;
.  -78,-12,0,6,-6,6,0,30,-24,6,0,30,-24,24,12,126,-78,-6,0,30,-24,24,6,114,-90,6,12,126,-54,102,72,450;
. -228,-60,0,6,-6,6,0...
(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
Index entries for sequences related to toothpick sequences

Cf. A011782, A048883, A139250, A139251, A160120, A160121, A151710, A173066, A173067, A173068, A173452, A173453.

More terms from Nathaniel Johnston, Nov 15 2010

A173453 a(n) = A160121(n) - A151710(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 78

Offset: 1

Omar E. Pol, May 29 2010

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. A048883, A139250, A139251, A160121, A151710, A173066, A173067, A173068, A173451, A173452.

A161326 Sequence to which the rows of the triangle of A160121 converge.

Original entry on oeis.org

9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 87, 33, 27, 51, 51, 57, 75, 129, 117, 75, 69, 117, 135

Offset: 1

Omar E. Pol, Jun 18 2009

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A153001, A147646, A160121.

A267701 a(n) = (A160121(n) - A266533(n))/6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 4, 0, 0, 0, 1, 0, 0, 1, 4, 0, 1, 1, 5, 2, 4, 4, 13, 2, 0, 0, 1, 0, 1, 1, 4, 0, 1, 1, 5, 2, 4, 4, 13, 2, 1, 1, 5, 2, 5, 6, 16, 5, 4, 4, 15, 8, 13, 15, 38, 10, 0, 0, 1, 0, 1

Offset: 1

Omar E. Pol, Jan 19 2016

a(n) is 1/6 of the difference between the number of Y-toothpicks added at n-th stage in the structure of A160120 and the number of Y-toothpicks added at n-th stage stage in the structure of A266532 (the outward version of A160120).

			Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
0;
0;
0,  0;
0,  0,0,0;
1,  0,0,0,1,0,1,1;
4,  0,0,0,1,0,0,1,4,0,1,1,5,2,4,4;
13, 2,0,0,1,0,1,1,4,0,1,1,5,2,4,4,13,2,1,1,5,2,5,6,16,5,4,4,15,8,13,15;
38,10,0,0,1,0,1...

Cf. A011782, A038573, A160120, A160121, A266532, A266533.

A160122 a(n) = A160121(n+1)*2/3.

Original entry on oeis.org

2, 2, 6, 2, 6, 6, 14, 6, 6, 6, 14, 10, 14, 18, 34, 18, 6, 6, 14, 10, 14, 18, 34, 22, 14, 18, 34, 34, 38, 46, 78, 54, 14, 6, 14, 10, 14, 18, 34, 22, 14, 18, 34, 34, 38, 46, 78, 58, 22, 18, 34, 34, 38, 50, 86, 78, 50, 46, 78, 90, 94, 114, 186, 154, 46, 6, 14

Offset: 1

Omar E. Pol, May 02 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

Y-toothpick sequence: A160120. Cf. A160121, A160123.

Cf. A139250, A139251. [From Omar E. Pol, Jun 16 2009]

More terms from Omar E. Pol, Jun 16 2009

A139251 First differences of toothpicks numbers A139250.

Original entry on oeis.org

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

A160120 Y-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 7, 16, 19, 28, 37, 58, 67, 76, 85, 106, 121, 142, 169, 220, 247, 256, 265, 286, 301, 322, 349, 400, 433, 454, 481, 532, 583, 640, 709, 826, 907, 928, 937, 958, 973, 994, 1021, 1072, 1105, 1126, 1153, 1204, 1255, 1312, 1381, 1498, 1585, 1618, 1645

Offset: 0

Omar E. Pol, May 02 2009

nonn

A Y-toothpick (or Y-shaped toothpick) is formed from three toothpicks of length 1, like a star with three endpoints and only one middle-point.
On the infinite triangular grid, we start at round 0 with no Y-toothpicks.
At round 1 we place a Y-toothpick anywhere in the plane.
At round 2 we add three more Y-toothpicks. After round 2, in the structure there are three rhombuses and a hexagon.
At round 3 we add three more Y-toothpicks.
And so on ... (see illustrations).
The sequence gives the number of Y-toothpicks after n rounds. A160121 (the first differences) gives the number added at the n-th round.
The Y-toothpick pattern has a recursive, fractal (or fractal-like) structure.
Note that, on the infinite triangular grid, a Y-toothpick can be represented as a polyedge with three components. In this case, at the n-th round, the structure is a polyedge with 3*a(n) components.
This structure is more complex than the toothpick structure of A139250. For example, at some rounds we can see inward growth.
The structure contains distinct polygons which have side length equal to 1.
Observation: It appears that the region of the structure where all grid points are covered is formed only by three distinct polygons:
- Triangles
- Rhombuses
- Concave-convex hexagons
Holes in the structure: Also, we can see distinct concave-convex polygons which contains a region where there are no grid points that are covered, for example:
- Decagons   (with  1 non-covered grid point)
- Dodecagons (with  4 non-covered grid points)
- 18-gons    (with  7 non-covered grid points)
- 30-gons    (with 26 non-covered grid points)
- ...
Observation: It appears that the number of distinct polygons that contain non-covered grid points is infinite.
This sequence appears to be related to powers of 2. For example:
Conjecture: It appears that if n = 2^k, k>0, then, between the other polygons, there appears a new centered hexagon formed by three rhombuses with side length = 2^k/2 = n/2.
Conjecture: Consider the perimeter of the structure. It appears that if n = 2^k, k>0, then the structure is a triangle-shaped polygon with A000225(k)*6 sides and a half toothpick in each vertice of the "triangle".
Conjecture: It appears that if n = 2^k, k>0, then the ratio of areas between the Y-toothpick structure and the unitary triangle is equal to A006516(k)*6.
See the entry A139250 for more information about the growth of "standard" toothpicks.
See also A160715 for another version of this structure but without internal growth of Y-toothpicks. [Omar E. Pol, May 31 2010]
For an alternative visualization replace every single toothpick with a rhombus, or in other words, replace every Y-toothpick with the "three-diamond" symbol, so we have a cellular automaton in which a(n) gives the total number of "three-diamond" symbols after n-th stage and A160167(n) counts the total number of "ON" diamonds in the structure after n-th stage. See also A253770. - Omar E. Pol, Dec 24 2015
The behavior is similar to A153006 (see the graph). - Omar E. Pol, Apr 03 2018

JungHwan Min, Table of n, a(n) for n = 0..5000
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.]
David Applegate, The movie version
Ayliean McDonald, Y-toothpick fractal, Twitter video (2021).
Olbaid Fractalium, Toothpick sequence Part 2, Youtube video (2023).
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the structure after 17 stages
Omar E. Pol, Illustration of the structure after 34 stages, from Applegate's movie version.
Omar E. Pol, Illustration: Fractal recursion, general step. (1)
Omar E. Pol, Illustration of initial terms of A139250, A160120, A147562 (Overlapping figures)
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Toothpick sequence: A139250.

Cf. A000079, A000225, A006516, A147562, A153006, A160121, A160123, A160715, A161206, A161328, A161330, A161430, A173066, A173068, A253770.

YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 0 && Count[out, 2] >= 2, 1, in == 0 && Count[out, 2] == 1, 2, True, in]]; A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* JungHwan Min, Jan 28 2016 *)
A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{435225738745686506433286166261571728070, 3, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* JungHwan Min, Jan 28 2016 *)

More terms from David Applegate, Jun 14 2009, Jun 18 2009

A147582 First differences of A147562.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 12, 36, 4, 12, 12, 36, 12, 36, 36, 108, 4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 4

Offset: 1

N. J. A. Sloane, Apr 29 2009

nonn

Bisection of A323651. - Omar E. Pol, Mar 04 2019

			From _Omar E. Pol_, Jun 14 2009: (Start)
When written as a triangle:
.1;
.4;
.4,12;
.4,12,12,36;
.4,12,12,36,12,36,36,108;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324,12,36,36,108,36,108,...
The rows converge to A161411. (End)

D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.
S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
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.]
David Applegate, The movie version
Omar E. Pol, Illustration of initial terms (Fig. 1: one-step rook), (Fig. 2: one-step bishop), (Fig. 3: overlapping squares), (Fig. 4: overlapping X-toothpicks), 2009
Omar E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures), 2009
D. Singmaster, On the cellular automaton of Ulam and Warburton, 2003 [Cached copy, included with permission]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021

Cf. A147562, A147610 (the sequence divided by 4), A048881, A000120.
Cf. A000079, A161411, A151779, A139250.
Cf. A048883, A139251, A160121, A162349. [Omar E. Pol, Nov 02 2009]
Cf. A323651.

A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120; A147582 := n-> if n <= 1 then n else 4*3^(wt(n-1)-1); fi; [seq(A147582(n),n=0..1000)]; # N. J. A. Sloane, Apr 07 2010

s = Plus @@ Flatten@ # & /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 200]; f[n_] = If[n == 0, 1, s[[n + 1]] - s[[n]]]; Array[f, 120, 0] (* Michael De Vlieger, Apr 09 2015, after Nadia Heninger and N. J. A. Sloane at A147562 *)

a(1) = 1; for n > 1, a(n) = 4*3^(wt(n-1)-1) where wt() = A000120(). - R. J. Mathar, Apr 30 2009
This formula is (essentially) given by Singmaster. - N. J. A. Sloane, Aug 06 2009
G.f.: x + 4*x*(Product_{k >= 0} (1 + 3*x^(2^k)) - 1)/3. - N. J. A. Sloane, Jun 10 2009

Extended by R. J. Mathar, Apr 30 2009

cp's OEIS Frontend

A160123 a(n) = A160121(n+1)/3.

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A173451 a(n) = A048883(n-1) - A160121(n).

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A173453 a(n) = A160121(n) - A151710(n).

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A161326 Sequence to which the rows of the triangle of A160121 converge.

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A267701 a(n) = (A160121(n) - A266533(n))/6.

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A160122 a(n) = A160121(n+1)*2/3.

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A139251 First differences of toothpicks numbers A139250.

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Maple

Mathematica

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A160120 Y-toothpick sequence (see Comments lines for definition).

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Mathematica

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A147582 First differences of A147562.

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Maple