A220494 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "wide" triangle of the first kind.

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

Offset: 0

Author

Omar E. Pol, Dec 23 2012

Keywords

• nonn

Comments

This cellular automaton uses toothpicks of length 1 and D-toothpicks of length 2^(1/2). Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints.

On the semi-infinite 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 D-toothpicks in the structure after n-th stage. A220495 (the first differences) give the number of toothpicks or D-toothpicks added at n-th 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

• 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.