A172311 -id:A172311 - cp's OEIS frontend

A172312 a(n) = A172311(n+1)/2.

1, 2, 3, 4, 6, 7, 7, 9, 9, 10, 12, 12, 19, 17, 21, 17, 13, 14, 16, 19, 25, 27, 29, 29, 33, 30, 28, 25, 33, 33, 33, 35, 21, 26, 28, 27, 31, 33, 39, 44

Offset: 1

Omar E. Pol, Jan 31 2010

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, A172310, A172311.

More terms from Nathaniel Johnston, Nov 15 2010

A172310 L-toothpick sequence (see Comment lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 13, 21, 33, 47, 61, 79, 97, 117, 141, 165, 203, 237, 279, 313, 339, 367, 399, 437, 489, 543, 607, 665, 733, 793, 853, 903, 969, 1039, 1109, 1183, 1233, 1285, 1345, 1399, 1463, 1529, 1613, 1701, 1817, 1923, 2055, 2155, 2291, 2417, 2557, 2663, 2781, 2881, 3003, 3109, 3247, 3361, 3499, 3631, 3783, 3939

Offset: 0

Omar E. Pol, Jan 31 2010

nonn

We define an "L-toothpick" to consist of two line segments forming an "L".
There are two size for L-toothpicks: Small and large. Each component of small L-toothpick has length 1. Each component of large L- toothpick has length sqrt(2).
The rule for the n-th stage:
If n is odd then we add the large L-toothpicks to the structure, otherwise we add the small L-toothpicks to the structure.
Note that, on the infinite square grid, every large L-toothpick is placed with angle = 45 degrees and every small L-toothpick is placed with angle = 90 degrees.
The special rule: L-toothpicks are not added if this would lead to overlap with another L-toothpick branch in the same generation.
We start at stage 0 with no L-toothpicks.
At stage 1 we place a large L-toothpick in the horizontal direction, as a "V", anywhere in the plane (Note that there are two exposed endpoints).
At stage 2 we place two small L-toothpicks.
At stage 3 we place four large L-toothpicks.
At stage 4 we place six small L-toothpicks.
And so on...
The sequence gives the number of L-toothpick after n stages. A172311 (the first differences) gives the number of L-toothpicks added at the n-th stage.
For more information see A139250, the toothpick sequence.
In calculating the extension, the "special rule" was strengthened to prohibit intersections as well as overlappings. [From John W. Layman, Feb 04 2010]
Note that the endpoints of the L-toothpicks of the new generation can touch the L-toothpìcks of old generations but the crosses and overlaps are prohibited. - Omar E. Pol, Mar 26 2016
The L-toothpick cellular automaton has an unusual property: the growths in its four wide wedges [North, East, South and West] have a recurrent behavior related to powers of 2, as we can find in other cellular automata (i.e., A194270). On the other hand, in its four narrow wedges [NE, SE, SW, NW] the behavior seems to be chaotic, without any recurrence, similar to the behavior of the snowflake cellular automaton of A161330. The remarkable fact is that with the same rules, different behaviors are produced. (See Applegate's movie version in the Links section.) - Omar E. Pol, Nov 06 2018

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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Omar E. Pol, Illustration of initial terms
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

For a similar version see A172304.
Cf. A161330 (snowflake).
Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172311, A172312, A194270, A220500.

Terms a(9)-a(41) from John W. Layman, Feb 04 2010

Corrected by David Applegate and Omar E. Pol; more terms beyond a(22) from David Applegate, Mar 26 2016

A194271 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194270.

Original entry on oeis.org

0, 1, 4, 8, 16, 22, 24, 22, 40, 40, 32, 32, 56, 74, 96, 50, 88, 72, 32, 48, 72, 104, 128, 112, 144, 144, 152, 96, 152, 178, 240, 122, 184, 136, 32, 48, 72, 108, 144, 144, 184, 188, 200, 176, 272, 274, 416, 250, 288, 272, 216, 144, 208, 292, 384, 332, 376

Offset: 0

Omar E. Pol, Aug 23 2011

nonn

Essentially the first differences of A194270.

			Written as a triangle:
0,
1,
4,
8,
16,22,
24,22,40,40,
32,32,56,74,96,50,88,72,
32,48,72,104,128,112,144,144,152,96,152,178,240,122,184,136,
32,48,72,108,144,144,184,188,200,176,272,274,416,250,288,...

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 toothpick sequences

Cf. A010694, A129370, A139250, A139251, A172311, A182839, A194270, A194441, A194443, A194445.

a(n) = n^2-(n-1)^2*(1-(-1)^n)/8, if 0 <= n <=4.
Let b(n) = A194441(n), let c(n) = A194443(n), let d(n) = A010694(n), then:
Conjecture: a(n) = 4*(b(n-1)-d(n)) + 2*(c(n)-d(n+1)) + 2*(c(n+2)-d(n+1)) + 8, if n >= 3.
Conjecture: a(2^k+2) = 32, if k >= 3.

More terms from Omar E. Pol, Sep 01 2011

A160173 Number of T-toothpicks added at n-th stage to the T-toothpick structure of A160172.

Original entry on oeis.org

0, 1, 3, 5, 9, 9, 9, 13, 25, 21, 9, 13, 25, 25, 25, 37, 73, 57, 9, 13, 25, 25, 25, 37, 73, 61, 25, 37, 73, 73, 73, 109, 217, 165, 9, 13, 25, 25, 25, 37, 73, 61, 25, 37, 73, 73, 73, 109, 217, 169, 25, 37, 73, 73, 73, 109, 217, 181, 73, 109, 217, 217, 217, 325, 649, 489, 9, 13, 25

Offset: 0

Omar E. Pol, Jun 01 2009

nonn

Essentially the first differences of A160172.
For further information see the Applegate-Pol-Sloane paper, chapter 11: T-shaped toothpicks. See also the figure 16 in the mentioned paper. - Omar E. Pol, Nov 18 2011
The numbers n in increasing order such that the triple [n, n, n] can be found here, give A199111. [Observed by Omar E. Pol, Nov 18 2011. Confirmed by Alois P. Heinz, Nov 21 2011]

			From _Omar E. Pol_, Feb 09 2010: (Start)
If written as a triangle:
0;
1;
3;
5;
9,9;
9,13,25,21;
9,13,25,25,25,37,73,57;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,165;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109,217,181,73,109,217,217,217,325,649,489;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109...
(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

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.], which is also available at arXiv:1004.3036v2
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A139251, A160172, A160121, A160171, A161207, A161329, A172311.

wt[n_] := DigitCount[n, 2, 1];
a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := 2/3 (3^wt[n-1] + 3^wt[n-2]) + 1;
Table[a[n], {n, 0, 68}] (* Jean-François Alcover, Aug 18 2018, after N. J. A. Sloane *)

a(n) = (2/3)*(3^wt(n-1) + 3^wt(n-2))+1 (where wt is A000120), for n >= 3. - N. J. A. Sloane, Jan 01 2010

More terms from N. J. A. Sloane, Jan 01 2010

A194445 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194444.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 11, 8, 4, 8, 16, 24, 12, 20, 25, 16, 4, 8, 16, 24, 28, 36, 42, 44, 20, 24, 40, 64, 32, 44, 53, 32, 4, 8, 16, 24, 28, 36, 44, 52, 42, 48, 60, 100, 68, 84, 83, 84, 28, 24, 44, 72, 84, 104, 116, 132, 54, 56, 92, 144, 72, 92, 109, 64, 4

Offset: 0

Omar E. Pol, Aug 24 2011

Essentially the first differences of A194444.

First differs from A220525 at a(13). - Omar E. Pol, Mar 23 2013

			Contribution from _Omar E. Pol_, Dec 05 2012 (Start):
Triangle begins:
0;
1;
2;
4,4;
4,8,11,8;
4,8,16,24,12,20,25,16;
4,8,16,24,28,36,42,44,20,24,40,64,32,44,53,32;
(End)

Row lengths give 1 together with A011782. Right border gives 0 together with A000079.

Cf. A139250, A139251, A172311, A182635, A182839, A194271, A194433, A194435, A194441, A194443, A194444, A212009.

It appears that a(2^k+1) = 4, if k >= 1.

a(n) = A194435(n)/4. - Omar E. Pol, Mar 23 2013

More terms from Omar E. Pol, Mar 23 2013

A172304 L-toothpick sequence starting with two opposite L-toothpicks.

Original entry on oeis.org

0, 2, 6, 14, 22, 30, 46, 62, 70, 86, 110, 134, 166, 190, 238, 278, 302, 318, 342, 382, 430, 470, 526, 582, 646, 710, 782, 838, 902, 950, 1030, 1118, 1150, 1182, 1246, 1318, 1382, 1422, 1486, 1566, 1662, 1766, 1910, 2006, 2134, 2254, 2414, 2526, 2622

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

The same as A172310 but starting with two L-toothpicks.
We start at stage 0 with no L-toothpicks.
At stage 1 we place two large L-toothpicks in the horizontal direction, as a "X", anywhere in the plane.
At stage 2 we place four small L-toothpicks.
At stage 3 we add eight more large L-toothpicks.
At stage 4 we add eight more small L-toothpicks.
And so on ...
The L-toothpick cellular automaton has an unusual property: the growths in its four wide wedges [North, East, South and West] have a recurrent behavior related to powers of 2, as we can find in other cellular automata (i.e., A212008). On the other hand, in its four narrow wedges [NE, SE, SW, NW] the behavior seems to be chaotic, without any recurrence, similar to the behavior of the snowflake cellular automaton of A161330. The remarkable fact is that with the same rules, different behaviors are produced. (See Applegate's movie version in the Links section.) - Omar E. Pol, Nov 06 2018

Yan Sheng Ang, Table of n, a(n) for n = 0..201
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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

For a similar version see A172310.
Cf. A161330 (snowflake).
Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172305, A172306, A172307, A172308, A172309, A172311, A172312, A172313, A212008, A220500.

Terms beyond a(14) from Yan Sheng Ang, Dec 10 2012

A172305 Number of L-toothpicks added to the L-toothpick structure of A172304 at the n-th stage.

Original entry on oeis.org

0, 2, 4, 8, 8, 8, 16, 16, 8, 16, 24, 24, 32, 24, 48, 40, 24, 16, 24, 40, 48, 40, 56, 56, 64, 64, 72, 56, 64, 48, 80, 88, 32, 32, 64, 72, 64, 40, 64, 80, 96, 104, 144, 96, 128, 120, 160, 112, 96, 80, 128, 128, 144, 88, 136, 128, 136, 144, 168, 168, 216, 160, 192, 168, 64, 80, 128

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

Robert Price, Table of n, a(n) for n = 0..202
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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. First differences of A172304.

Cf. A139250, A139251, A172304, A172306, A172307, A172308, A172309, A172310, A172311, A172312, A172313.

More terms from Colin Barker, Apr 19 2015

a(49)-a(66) from Robert Price, Jun 17 2019

A172306 a(n) = A172304(n)/2.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 31, 35, 43, 55, 67, 83, 95, 119, 139, 151, 159, 171, 191, 215, 235, 263, 291, 323, 355, 391, 419, 451, 475, 515, 559, 575, 591, 623, 659, 691, 711, 743, 783, 831, 883, 955, 1003, 1067, 1127, 1207, 1263, 1311, 1351, 1415, 1479, 1551, 1595, 1663, 1727, 1795, 1867, 1951, 2035, 2143, 2223, 2319, 2403, 2435, 2475, 2539, 2587

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

Robert Price, Table of n, a(n) for n = 0..202
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, A172304, A172305, A172307, A172308, A172309, A172310, A172311, A172312, A172313.

a(14)-a(67) from Robert Price, Jun 17 2019

A172307 a(n) = A172305(n)/2.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 8, 4, 8, 12, 12, 16, 12, 24, 20, 12, 8, 12, 20, 24, 20, 28, 28, 32, 32, 36, 28, 32, 24, 40, 44, 16, 16, 32, 36, 32, 20, 32, 40, 48, 52, 72, 48, 64, 60, 80, 56, 48, 40, 64, 64, 72, 44, 68, 64, 68, 72, 84, 84, 108, 80, 96, 84, 32, 40, 64

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

Robert Price, Table of n, a(n) for n = 0..202
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, A172304, A172305, A172306, A172308, A172309, A172310, A172311, A172312, A172313.

a(14)-a(66) from Robert Price, Jun 17 2019

A172308 L-toothpick sequence in the first quadrant.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 17, 21, 27, 33, 41, 47, 59, 69, 75, 79, 85, 95, 107, 117, 131, 145, 161, 177, 195, 209, 225, 237, 257, 279, 287, 295, 311, 329, 345, 355, 371, 391, 415, 441, 477, 501, 533, 563, 603, 631, 655

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

The same as A172310 and A172304, but starting from half L-toothpick in the first quadrant.
Note that if n is odd then we add the small L-toothpicks to the structure, otherwise we add the large L-toothpicks to the structure.
We start at stage 0 with half L-toothpick: A segment from (0,0) to (1,1).
At stage 1 we place a small L-toothpick at the exposed toothpick end.
At stage 2 we place two large L-toothpicks.
At stage 3 we place two small L-toothpicks.
At stage 4 we place two large L-toothpicks.
And so on...
The sequence gives the number of L-toothpicks after n stages. A172309 (the first differences) gives the number of L-toothpicks added at the n-th stage.

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, A153000, A160120, A160170, A160172, A161206, A161328, A172304, A172305, A172306, A172307, A172309, A172310, A172311, A172312, A172313.

a(17)-a(47) from Robert Price, Jun 17 2019

cp's OEIS Frontend

A172312 a(n) = A172311(n+1)/2.

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A172310 L-toothpick sequence (see Comment lines for definition).

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A194271 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194270.

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A160173 Number of T-toothpicks added at n-th stage to the T-toothpick structure of A160172.

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A194445 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194444.

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A172304 L-toothpick sequence starting with two opposite L-toothpicks.

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A172305 Number of L-toothpicks added to the L-toothpick structure of A172304 at the n-th stage.

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A172306 a(n) = A172304(n)/2.

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A172307 a(n) = A172305(n)/2.

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A172308 L-toothpick sequence in the first quadrant.

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