

A172310


Ltoothpick sequence (see Comment lines for definition).


21



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
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OFFSET

0,3


COMMENTS

We define an "Ltoothpick" to consist of two line segments forming an "L".
There are two size for Ltoothpicks: Small and large. Each component of small Ltoothpick has length 1. Each component of large L toothpick has length sqrt(2).
The rule for the nth stage:
If n is odd then we add the large Ltoothpicks to the structure, otherwise we add the small Ltoothpicks to the structure.
Note that, on the infinite square grid, every large Ltoothpick is placed with angle = 45 degrees and every small Ltoothpick is placed with angle = 90 degrees.
The special rule: Ltoothpicks are not added if this would lead to overlap with another Ltoothpick branch in the same generation.
We start at stage 0 with no Ltoothpicks.
At stage 1 we place a large Ltoothpick 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 Ltoothpicks.
At stage 3 we place four large Ltoothpicks.
At stage 4 we place six small Ltoothpicks.
And so on...
The sequence gives the number of Ltoothpick after n stages. A172311 (the first differences) gives the number of Ltoothpicks added at the nth 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 Ltoothpicks of the new generation can touch the Ltoothpìcks of old generations but the crosses and overlaps are prohibited.  Omar E. Pol, Mar 26 2016
The Ltoothpick 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


LINKS

Table of n, a(n) for n=0..59.
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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)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


CROSSREFS

For a similar version see A172304.
Cf. A161330 (snowflake).
Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172311, A172312, A194270, A220500.
Sequence in context: A004136 A147409 A147342 * A060939 A174030 A098575
Adjacent sequences: A172307 A172308 A172309 * A172311 A172312 A172313


KEYWORD

nonn


AUTHOR

Omar E. Pol, Jan 31 2010


EXTENSIONS

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


STATUS

approved



