

A212008


Dtoothpick sequence of the second kind starting with a single toothpick.


4



0, 1, 5, 13, 29, 51, 71, 95, 131, 171, 203, 247, 303, 397, 457, 513, 589, 661, 693, 741, 813, 925, 1057, 1197, 1333, 1501, 1613, 1745, 1885, 2123, 2271, 2391, 2547, 2683, 2715, 2763, 2835, 2947, 3079
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OFFSET

0,3


COMMENTS

This cellular automaton uses elements of two sizes: toothpicks of length 1 and Dtoothpicks of length 2^(1/2). Toothpicks are placed in horizontal or vertical direction. Dtoothpicks are placed in diagonal direction. Toothpicks and Dtoothpicks are connected by their endpoints.
On the infinite square grid we start with no elements.
At stage 1, place a single toothpick on the paper, aligned with the yaxis.
The rule for adding new elements is as follows. If it is possible, each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees, otherwise each exposed endpoint of the elements of the old generation must be touched by an endpoint of an element of the new generation such that the angle between the old element and the new element is equal to 135 degrees. Intersections and overlapping are prohibited. The sequence gives the number of toothpicks and Dtoothpicks in the structure after nth stage. The first differences (A212009) give the number of toothpicks or Dtoothpicks added at nth stage.
It appears that if n >> 1 the structure looks like an octagon. This C.A. has a fractal (or likefractal) behavior related to powers of 2. Note that for some values of n we can see an internal growth.
The structure contains eight wedges. Each vertical wedge also contains infinitely many copies of the oblique wedges. Each oblique wedge also contains infinitely many copies of the vertical wedges. Finally, each horizontal wedge also contains infinitely many copies of the vertical wedges and of the oblique wedges.
The structure appears to be a puzzle which contains at least 50 distinct internal regions (or polygonal pieces), and possibly more. Some of them appear for first time after 200 stages. The largest known polygon is a concave 24gon.
Also the structure contains infinitely many copies of two subsets of distinct size which are formed by five polygons: three hexagons, a 9gon and a pentagon. The distribution of these subsets have a surprising connection with the Sierpinski triangle A047999, but here the pattern is more complex.
For another version see A220500.


LINKS

Table of n, a(n) for n=0..38.
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


CROSSREFS

Cf. A047999, A139250, A194270, A194432, A194434, A194440, A194442, A194444, A220500.
Sequence in context: A272801 A100438 A129371 * A194270 A194700 A220500
Adjacent sequences: A212005 A212006 A212007 * A212009 A212010 A212011


KEYWORD

nonn,more


AUTHOR

Omar E. Pol, Dec 15 2012


STATUS

approved



