

A172310


Ltoothpick sequence (see Comment lines for definition).


22



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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



CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



