A160172 T-toothpick sequence (see Comments lines for definition).

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

Offset: 0

Author

Omar E. Pol, Jun 01 2009

Keywords

• nonn
• nice

Comments

A T-toothpick 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 T-toothpicks.

At round 1 we place a T-toothpick anywhere in the plane.

At round 2 we place three other T-toothpicks.

And so on...

The rule for adding a new T-toothpick is the following. A new T-toothpick 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 T-toothpicks after n rounds. A160173 (the first differences) gives the number added at the n-th round.

See the entry A139250 for more information about the toothpick process and the toothpick propagation.

On the infinite square grid a T-toothpick can be represented as a square polyedge with three components from a central point: two consecutive components on the same straight-line and a centered orthogonal component.

If the T-toothpick has three components then at the n-th 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 Applegate-Pol-Sloane paper, chapter 11, "T-shaped toothpicks". See also A160173.

Also, this sequence can be illustrated using another structure in which every T-toothpick 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 Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2015

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

Crossrefs

Cf. A139250, A139251, A147562, A160120, A160160, A160164, A160170, A160173, A160406, A160408, A160426, A160800, A162795, A169707, A187220, A255366, A256260.

Programs

Mathematica

wt[n_] := DigitCount[n, 2, 1];
A151920[n_] := Sum[3^wt[i], {i, 1, n + 1}]/3;
a[n_] := 2*A151920[n - 2] + 2*A151920[n - 3] + n;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 21 2024, after Charlie Neder *)

Formula

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

Extensions

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