A160121 -id:A160121 - cp's OEIS frontend

A159913 a(n) = 2^(A000120(n) + 1) - 1, where A000120(n) = number of nonzero bits in n.

1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 7, 15, 15, 31, 15, 31, 31, 63, 15, 31, 31, 63, 31, 63, 63, 127, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31

Offset: 0

M. F. Hasler, May 03 2009

nonn

Essentially the same sequence as A117973 and A001316. The latter entry has much more information. - N. J. A. Sloane, Jun 05 2009
First differences of A159912; every other term of A038573.
Equals Sierpinski's gasket, A047999; as an infinite lower triangular matrix * [1,2,2,2,...] as a vector. - Gary W. Adamson, Oct 16 2009
a(n) is also the number of cells turned ON at n-th generation in the outward corner version of the Ulam-Warburton cellular automaton of A147562, and a(n) is also the number of Y-toothpicks added at n-th generation in the outward corner version of the Y-toothpick structure of A160120. - David Applegate and Omar E. Pol, Jan 24 2016

			From _Michael De Vlieger_, Jan 25 2016: (Start)
The number n converted to binary, "0" represented by "." for better visibility of 1's, totaling the 1's and calculating the sequence:
n    Binary   Total                         a(n)
0 -> .     ->     0, thus 2^(0+1)-1 =  2-1 =  1
1 -> 1     ->     1,   "  2^(1+1)-1 =  4-1 =  3
2 -> 1.    ->     1,   "  2^(1+1)-1 =  4-1 =  3
3 -> 11    ->     2,   "  2^(2+1)-1 =  8-1 =  7
4 -> 1..   ->     1,   "  2^(1+1)-1 =  4-1 =  3
5 -> 1.1   ->     2,   "  2^(2+1)-1 =  8-1 =  7
6 -> 11.   ->     2,   "  2^(2+1)-1 =  8-1 =  7
7 -> 111   ->     3,   "  2^(3+1)-1 = 16-1 = 15
8 -> 1...  ->     1,   "  2^(1+1)-1 =  4-1 =  3
9 -> 1..1  ->     2,   "  2^(2+1)-1 =  8-1 =  7
10-> 1.1.  ->     2,   "  2^(2+1)-1 =  8-1 =  7
(End)

David Applegate, The movie version
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

Rows of triangle in A038573 converge to this sequence. - N. J. A. Sloane, Jun 05 2009

Cf. A000120, A038573, A047999, A159912, A117973, A001316, A147582, A160121.

Table[2^(DigitCount[n, 2][[1]] + 1) - 1, {n, 0, 78}] (* or *)
Table[2^(Total@ IntegerDigits[n, 2] + 1) - 1, {n, 0, 78}] (* Michael De Vlieger, Jan 25 2016 *)

PARI
```
A159913(n)=2<
				
```

def A159913(n): return (1<Chai Wah Wu, Nov 15 2022

a(n) = 2^A000120(2n+1) - 1 = A038573(2n+1) = 2*A038573(n) + 1 = A159912(n+1) - A159912(n).
a(n) = A160019(n,n). - Philippe Deléham, Nov 15 2011
a(n) = n - Sum_{k=0..n} (-1)^binomial(n, k). - Peter Luschny, Jan 14 2018

A161431 First differences of A161430.

Original entry on oeis.org

9, 24, 24, 66, 24

Offset: 1

Omar E. Pol, Jun 17 2009

Cf. A139251, A160121, A161430, A161432, A161433.

A253770 Number of ON states after n generations of cellular automaton based on triangles, with diamonds.

Original entry on oeis.org

0, 6, 24, 42, 96, 114, 168, 222, 348, 402, 456, 510, 636, 726, 852, 1014, 1320, 1482, 1536, 1590, 1716, 1806, 1932, 2094, 2400, 2598, 2724, 2886, 3192, 3498, 3840, 4254, 4956, 5442, 5568, 5622, 5748, 5838, 5964, 6126, 6432, 6630, 6756, 6918, 7224, 7530, 7872, 8286

Offset: 0

Omar E. Pol, Jan 11 2015

nonn

Also 6 times the Y-toothpicks sequence A160120.
Explanation: consider the Y-toothpick structure of A160120, then replace every Y-toothpick with six ON cells forming a star with three rhombuses (or diamonds) that share only one vertex. Every diamond contains two triangular cells that share one edge.
The rules are the essentially the same as A160120.
An ON cell remains ON forever.
The sequence gives the number of triangular ON cells after the n-th stage.
A253771 (the first differences) give the number of triangular cells turned "ON" at the n-th stage.
A160120 (the Y-toothpick sequence) gives the number of stars in the structure after the n-th stage.
A160121 gives the number of stars added at the n-th stage.
A160167 gives the number of diamonds in the structure after the n-th stage.

			After one generation, the cellular automaton looks like a star or a flower with three petals as shown below:
.
.        /\
.       _\/_
.      /_/\_\
.
There are one star, three diamonds and six ON cells, so a(1) = 6.

Cf. A139250, A147562, A151723, A160120, A160157, A160167, A161644, A182632, A250300, A253771.

a(n) = 6*A160120(n) = 3*A160157(n) = 2*A160167(n).

A327331 Number of elements added at n-th stage to the toothpick structure of A327330.

Original entry on oeis.org

1, 2, 4, 4, 4, 8, 10, 8, 4, 8, 10, 12, 14, 22, 22, 16, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 36, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 40, 18, 28, 34, 46, 50, 58, 50, 48, 40, 68, 76, 84, 108, 156, 100, 76, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 40

Offset: 1

Omar E. Pol, Sep 01 2019

The word of this cellular automaton is "ab".
The structure of the irregular triangle is as shown below:
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;
...
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
Columns "a" contain numbers of I-toothpicks.
Columns "b" contain numbers of V-toothpicks.
For further information about the word of cellular automata see A296612.

			Triangle begins:
1,2;
4,4;
4,8,10,8;
4,8,10,12,14,22,22,16;
4,8,10,12,14,22,22,20,14,24,28,34,42,60,48,36;
4,8,10,12,14,22,22,20,14,24,28,34,42,60,48,40,18,28,34,46,50,58,50,48,40,68,...

First differences of A327330.
Column 1 gives A123932.
First differs from A231348 at a(11).
Cf. A011782, A139250, A139251, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649, A327333 (similar triangle).
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323647, A323651.

A327333 Number of elements added at n-th stage to the toothpick structure of A327332.

Original entry on oeis.org

1, 2, 4, 4, 4, 6, 12, 8, 4, 6, 12, 12, 10, 16, 32, 16, 4, 6, 12, 12, 10, 16, 32, 20, 12, 18, 36, 36, 26, 42, 84, 32, 4, 6, 12, 12, 10, 16, 32, 20, 12, 18, 36, 36, 26, 42, 84, 40, 16, 24, 48, 44, 24, 40, 80, 48, 32, 48, 96, 96, 64, 104, 208, 64, 4, 6, 12, 12, 10, 16, 32, 20, 12, 18, 36, 36, 26, 42, 84, 40

Offset: 1

Omar E. Pol, Sep 01 2019

The word of this cellular automaton is "ab".
The structure of the irregular triangle is as shown below:
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;
...
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
Columns "a" contain numbers of V-toothpicks. Columns "b" contain numbers of I-toothpicks. See the example.
For further information about the word of cellular automata see A296612.

			Triangle begins:
1,2;
4,4;
4,6,12,8;
4,6,12,12,10,16,32,16;
4,6,12,12,10,16,32,20,12,18,36,36,26,42,84,32;
4,6,12,12,10,16,32,20,12,18,36,36,26,42,84,40,16,24,48,44,24,40,80,48,32,48,...
It appears that right border gives the even powers of 2.

First differences of A327332.
Column 1 gives A123932.
Cf. A011782, A139250, A139251, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649, A327331 (similar triangle).
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323647, A323651.

A161833 First differences of A161832.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 2, 2, 2, 4, 3, 4, 5, 9, 5, 2

Offset: 1

Omar E. Pol, Jun 20 2009

Number of Y-toothpicks added to the sieve at the n-th round.

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. A139251, A160121, A160407, A161830, A161831, A161832.

a(n) = A161831(n+1)/2.

A220498 Number of E-toothpicks (or tridents) added at n-th stage to the structure of the equilateral triangle of A220478.

Original entry on oeis.org

0, 2, 2, 2, 4, 2, 4, 4, 4, 6, 4, 6, 8, 2, 4, 4, 6, 10, 6, 14, 8, 10, 14, 8, 12, 14, 4, 8, 8, 10, 16, 12, 22, 16, 16, 18, 12, 14, 16, 16, 16, 10, 12, 20, 14, 22, 22, 18, 18, 24, 18, 28, 18, 20, 28, 22, 28, 20, 18, 18, 22, 32, 32, 26, 24, 22, 28, 28, 32, 34, 20, 20, 28

Offset: 0

Omar E. Pol, Feb 19 2013

nonn

Essentially the first differences of A220478.

Cf. A139250, A139251, A160121, A161330, A161329, A161331, A211974, A211976.

a(n) = 1 + A161331(n+1)/6 = 2*A211976(n).

A160425 a(n) = number of grid points that are covered after n-th rounds of A160120.

Original entry on oeis.org

0, 4, 10, 19, 31, 40

Offset: 0

Omar E. Pol, Jun 01 2009

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, A147614, A160120, A160121, A160128.

A161413 First differences of A161412.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 3, 4, 6, 4, 5, 5, 10, 9, 5

Offset: 1

Omar E. Pol, Jun 10 2009

Number of V-toothpicks added to the structure at the n-th round.

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. A139251, A160121, A160407, A161207, A161329, A161412.

A161427 First differences of A161426.

Original entry on oeis.org

1, 3, 3, 7, 5, 7, 9, 17, 11, 7

Offset: 1

Omar E. Pol and David Applegate, Jun 20 2009

Number of Y-toothpicks added to the sieve at the n-th round.

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
David Applegate, The movie version

Cf. A139251, A152980, A160121, A161426, A161831.

cp's OEIS Frontend

A159913 a(n) = 2^(A000120(n) + 1) - 1, where A000120(n) = number of nonzero bits in n.

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A161431 First differences of A161430.

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A253770 Number of ON states after n generations of cellular automaton based on triangles, with diamonds.

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A327331 Number of elements added at n-th stage to the toothpick structure of A327330.

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A327333 Number of elements added at n-th stage to the toothpick structure of A327332.

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A161833 First differences of A161832.

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A220498 Number of E-toothpicks (or tridents) added at n-th stage to the structure of the equilateral triangle of A220478.

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A160425 a(n) = number of grid points that are covered after n-th rounds of A160120.

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A161413 First differences of A161412.

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A161427 First differences of A161426.

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