A172310 L-toothpick sequence (see Comment lines for definition).

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

Offset: 0

Author

Omar E. Pol, Jan 31 2010

Keywords

• nonn

Comments

We define an "L-toothpick" to consist of two line segments forming an "L".

There are two size for L-toothpicks: Small and large. Each component of small L-toothpick has length 1. Each component of large L- toothpick has length sqrt(2).

The rule for the n-th stage:

If n is odd then we add the large L-toothpicks to the structure, otherwise we add the small L-toothpicks to the structure.

Note that, on the infinite square grid, every large L-toothpick is placed with angle = 45 degrees and every small L-toothpick is placed with angle = 90 degrees.

The special rule: L-toothpicks are not added if this would lead to overlap with another L-toothpick branch in the same generation.

We start at stage 0 with no L-toothpicks.

At stage 1 we place a large L-toothpick 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 L-toothpicks.

At stage 3 we place four large L-toothpicks.

At stage 4 we place six small L-toothpicks.

And so on...

The sequence gives the number of L-toothpick after n stages. A172311 (the first differences) gives the number of L-toothpicks added at the n-th 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 L-toothpicks of the new generation can touch the L-toothpìcks of old generations but the crosses and overlaps are prohibited. - Omar E. Pol, Mar 26 2016

The L-toothpick 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

• 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), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-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.

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