

A160172


Ttoothpick sequence (see Comments lines for definition).


19



0, 1, 4, 9, 18, 27, 36, 49, 74, 95, 104, 117, 142, 167, 192, 229, 302, 359, 368, 381, 406, 431, 456, 493, 566, 627, 652, 689, 762, 835, 908, 1017, 1234, 1399, 1408, 1421, 1446, 1471, 1496, 1533, 1606, 1667, 1692, 1729, 1802, 1875, 1948, 2057, 2274, 2443, 2468
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OFFSET

0,3


COMMENTS

A Ttoothpick is formed from three toothpicks of equal length, in the shape of a T. There are three endpoints. We call the middle of the top toothpick the pivot point.
We start at round 0 with no Ttoothpicks.
At round 1 we place a Ttoothpick anywhere in the plane.
At round 2 we place three other Ttoothpicks.
And so on...
The rule for adding a new Ttoothpick is the following. A new Ttoothpick is added at any exposed endpoint, with the pivot point touching the endpoint and so that the crossbar of the new toothpick is perpendicular to the exposed end.
The sequence gives the number of Ttoothpicks after n rounds. A160173 (the first differences) gives the number added at the nth round.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
On the infinite square grid a Ttoothpick can be represented as a square polyedge with three components from a central point: two consecutive components on the same straightline and a centered orthogonal component.
If the Ttoothpick has three components then at the nth round the structure is a polyedge with 3*a(n) components.
From Omar E. Pol, Mar 26 2011: (Start)
For formula and more information see the ApplegatePolSloane paper, chapter 11, "Tshaped toothpicks". See also A160173.
Also, this sequence can be illustrated using another structure in which every Ttoothpick is replaced by an isosceles right triangle. (End)
The structure is very distinct but the graph is similar to the graphs from the following sequences: A147562, A160164, A162795, A169707, A187220, A255366, A256260, at least for the known terms from Data section.  Omar E. Pol, Nov 24 2015
Shares with A255366 some terms with the same index, for example the element a(43) = 1729, the HardyRamanujan number.  Omar E. Pol, Nov 25 2015


LINKS

Table of n, a(n) for n=0..50.
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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.],
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS


FORMULA

a(n) = 2*A151920(n) + 2*A151920(n1) + n + 1.  Charlie Neder, Feb 07 2019


CROSSREFS

Cf. A139250, A139251, A147562, A160120, A160160, A160164, A160170, A160173, A160406, A160408, A160426, A160800, A162795, A169707, A187220, A255366, A256260.
Sequence in context: A062952 A229027 A100435 * A256536 A026412 A101424
Adjacent sequences: A160169 A160170 A160171 * A160173 A160174 A160175


KEYWORD

nonn,nice


AUTHOR

Omar E. Pol, Jun 01 2009


EXTENSIONS

Edited and extended by N. J. A. Sloane, Jan 01 2010


STATUS

approved



