A152980 -id:A152980 - cp's OEIS frontend

A194696 a(n) = 2*A194693(n).

4, 4, 8, 12, 16, 16, 24, 26, 24, 20, 32, 40, 64, 40, 48, 54, 40, 20, 32, 48, 72, 82, 96, 96, 108, 68, 88, 100, 160, 96, 96, 110, 72, 20, 32, 48, 72, 82, 96, 108, 136, 124, 160, 160, 248, 190, 208, 178, 188, 88, 96, 136

Offset: 1

Omar E. Pol, Sep 03 2011

Conjecture: number of toothpicks or D-toothpicks added to the structure of A194440 at stage 2^k+n, if k tends to infinity. It appears that rows of A194441 when written as a triangle converge to this sequence.

			Written as a triangle:
4,
4,
8,
12,16,
16,24,26,24,
20,32,40,64,40,48,54,40,
20,32,48,72,82,96,96,108,68,88,100,160,96,96,110,72,
20,32,48,72,82,96,108,136,124,160,160,248,190,208,178,...

Cf. A152980, A194270, A194440, A194692, A194693, A194697.

A194697 a(n) = 2*A194695(n).

Original entry on oeis.org

4, 4, 8, 12, 16, 16, 22, 32, 26, 20, 24, 40, 44, 52, 48, 72, 42, 20, 24

Offset: 1

Omar E. Pol, Sep 03 2011

Conjecture: number of toothpicks or D-toothpicks added to the structure of A194442 at stage 2^k+n, if k tends to infinity. It appears that rows of A194443 when written as a triangle converge to this sequence.

			Written as a triangle:
4,
4,
8,
12,16,
16,22,32,26,
20,24,40,44,52,48,72,42,
20,24

Cf. A152980, A194270, A194442, A194694, A194695, A194696

A160416 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 8, 11, 32, 39, 80, 89, 146, 159

Offset: 0

Omar E. Pol, May 20 2009, Jun 14 2009

nonn

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
..9...9...9...9...9
...888.888.888.888.
...878.878.878.878.
...886668666866688.
..9..656.656.656..9
...886644464446688.
...878.434.434.878.
...886644222446688.
..9..656.212.656..9
000000000022446688.
0000000000.434.878.
000000000064446688.
000000000056.656..9
000000000066866688.
0000000000.878.878.
0000000000.888.888.
00000000009...9...9
0000000000.........
0000000000.........

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, A152980, A153006, A160117, A160118, A160410, A160412, A160414.

Cf. A160796, A161417. [From Omar E. Pol, Jun 14 2009]

A161212 a(n) = A161210(n)*2.

Original entry on oeis.org

0, 2, 6, 14, 28, 42, 56

Offset: 0

Omar E. Pol, Jun 06 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, A152980, A153006, A160160, A161210, A161213, A161214, A161216.

A161214 a(n) = A161210(n)*3.

Original entry on oeis.org

0, 3, 9, 21, 42, 63, 84

Offset: 0

Omar E. Pol, Jun 06 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, A152980, A153006, A160160, A161210, A161212, A161215, A161216.

A161216 a(n) = A161210(n)*4.

Original entry on oeis.org

0, 4, 12, 28, 56, 84, 112

Offset: 0

Omar E. Pol, Jun 06 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, A152980, A153006, A160160, A161210, A161212, A161214, A161217.

A162349 First differences of A160412.

Original entry on oeis.org

3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 27, 81, 81, 243, 81, 243, 243, 729, 81, 243, 243, 729, 243, 729

Offset: 1

Omar E. Pol, Jul 14 2009

nonn

Note that if A048883 is written as a triangle then rows converge to this sequence. - Omar E. Pol, Nov 15 2009

Omar E. Pol, Illustration of initial terms (Neighbors of the vertices).

Cf. A063787, A152980, A160410, A160411, A160412, A160414, A160415, A160417, A160797.

Cf. A048883, A161415. - Omar E. Pol, Nov 15 2009

a[n_] := 3^(1 + DigitCount[n - 1, 2, 1]); Array[a, 100] (* Amiram Eldar, Feb 02 2024 *)

a(n) = 3^A063787(n) = 3 * A048883(n-1). - Amiram Eldar, Feb 02 2024

More terms from Omar E. Pol, Nov 15 2009
More terms from Colin Barker, Apr 19 2015
More terms from Amiram Eldar, Feb 02 2024

A168113 First differences of A168112.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 6, 7, 6, 3, 6, 7, 8, 9, 16, 17, 10, 3, 6, 7, 8, 9, 16, 17, 12, 9, 16, 19, 22, 31, 46, 41, 18, 3, 6, 7, 8, 9, 16, 17, 12, 9, 16, 19, 22, 31, 46, 41, 20, 9, 16, 19, 22, 31, 46, 43, 30, 31, 48

Offset: 1

Omar E. Pol, Dec 07 2009

a(n) is the number of toothpicks added at the n-th round in the toothpick structure of A168112.

			If written as a triangle, begins:
1;
1;
2;
3,3;
3,6,7,6;
3,6,7,8,9,16,17,10;
3,6,7,8,9,16,17,12,9,16,19,22,31,46,41,18;
Rows converge to A168114.

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, A152978, A152980, A160407, A168112, A168114.

Observation: It appears that a(2^i+2) = 3, for i>0.

a(n) = A160731(n)/2. [From Nathaniel Johnston, Mar 28 2011]

Terms after a(34) from Nathaniel Johnston, Mar 28 2011

A152979 Records in A152978.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 10, 15, 22, 23, 28, 35, 52, 64, 65, 74, 91, 122, 168, 176, 177, 186, 239, 304, 412, 512, 513, 524, 527, 599, 782, 1020, 1336, 1488, 1489, 1500, 1575, 1980, 2584, 3376, 4160, 4161, 4172, 4174, 4247, 4275, 4653, 5010, 6544, 8544

Offset: 1

Omar E. Pol, Dec 23 2008

nonn

This sequence also gives the records in A152980. - Nathaniel Johnston, Apr 10 2011

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. A039251, A152768, A152968, A152973, A152978, A152998, A153000, A153006. Toothpick sequence: A139250.

a(16)-a(51) from Nathaniel Johnston, Apr 10 2011

A159793 a(n) = A153006(n)*2.

Original entry on oeis.org

0, 2, 6, 12, 18, 26, 40, 56, 66, 74, 88, 106, 126, 156, 200, 240, 258, 266, 280, 298, 318, 348, 392, 434, 462, 492, 538, 594, 664, 768, 896, 992, 1026, 1034, 1048, 1066, 1086, 1116, 1160, 1202, 1230, 1260, 1306, 1362, 1432, 1536, 1664, 1762, 1806, 1836, 1882, 1938

Offset: 0

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

Toothpick sequence: A139250.

Cf. A139251, A152980, A153006, A159794, A159795.

lista(nn) = concat(0, 2*Vec(x*((1+x)/(1-x))*prod(k=1, nn, 1+x^(2^k-1)+2*x^(2^k)) + O(x^nn))); \\ Jinyuan Wang, Mar 04 2020

More terms from Jinyuan Wang, Mar 04 2020

cp's OEIS Frontend

A194696 a(n) = 2*A194693(n).

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A194697 a(n) = 2*A194695(n).

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A160416 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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A161212 a(n) = A161210(n)*2.

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A161214 a(n) = A161210(n)*3.

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A161216 a(n) = A161210(n)*4.

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A162349 First differences of A160412.

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A168113 First differences of A168112.

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A152979 Records in A152978.

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A159793 a(n) = A153006(n)*2.

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