

A220494


Number of toothpicks and Dtoothpicks after nth stage in the structure of the Dtoothpick "wide" triangle of the first kind.


4



0, 1, 3, 7, 11, 15, 19, 27, 35, 39, 43, 51, 59, 67, 75, 91, 107, 111, 115, 123, 131, 139, 147, 163, 179, 187, 195, 211, 227, 243, 259, 291, 323, 327, 331, 339, 347, 355, 363, 379, 395, 403, 411, 427, 443, 459, 475, 507, 539, 547, 555, 571, 587, 603
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OFFSET

0,3


COMMENTS

This cellular automaton uses 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 semiinfinite square grid we start with no elements, so a(0) = 0. At stage 1, we place a single toothpick in vertical direction at (0,0),(0,1), so a(1) = 1. Note that there is only one exposed toothpick endpoint.
The rules for adding new elements are as follows. 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. The endpoints of the toothpicks of the old generation that are perpendiculars to the initial toothpick remain exposed forever. Overlapping is prohibited.
The sequence gives the number of toothpicks and Dtoothpicks in the structure after nth stage. A220495 (the first differences) give the number of toothpicks or Dtoothpicks added at nth stage.
It appears that the structure has fractal behavior related to powers of 2. It appears that this cellular automaton has a surprising connection with the Sierpinski triangle, but here the structure is more complex.
For a similar version see A220496. For other more complex versions see A194440, A220520. First differs from A194440 (and from A220520) at a(12).


LINKS

Table of n, a(n) for n=0..53.
Omar E. Pol, Illustration of the structure after 32 stages
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, A194440, A220495, A220496, A220520.
Sequence in context: A220526 A034934 A191151 * A194440 A220520 A330165
Adjacent sequences: A220491 A220492 A220493 * A220495 A220496 A220497


KEYWORD

nonn


AUTHOR

Omar E. Pol, Dec 23 2012


STATUS

approved



