A327332 "Concave pentagon" toothpick sequence, starting with a V-toothpick (see Comments for precise definition).

0, 1, 3, 7, 11, 15, 21, 33, 41, 45, 51, 63, 75, 85, 101, 133, 149, 153, 159, 171, 183, 193, 209, 241, 261, 273, 291, 327, 363, 389, 431, 515, 547, 551, 557, 569, 581, 591, 607, 639, 659, 671, 689, 725, 761, 787, 829, 913, 953, 969, 993, 1041, 1085, 1109, 1149, 1229, 1277, 1309, 1357, 1453, 1549, 1613

Offset: 0

Author

Omar E. Pol, Sep 01 2019

Keywords

• nonn

Comments

Another version and very similar to A327330.

This arises from a hybrid cellular automaton on a triangular grid formed of V-toothpicks (A161206) and I-toothpicks (A160164).

After 2^k stages, the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two right triangles (R1 and R2) both with their hypotenuses in vertical position, as shown below:

.

* *

* * * *

* * * *

* * *

* R1 * * R2 *

* * * *

* * * *

* * * *

* * E * *

* * * *

* * * *

** **

* * * * * * * * * *

.

Every triangle has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.

For the construction of the sequence the rules are as follows:

On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.

At stage 1 we place an V-toothpick, formed of two single toothpicks, with its central vertice directed up, like a gable roof, so a(1) = 1.

For the next n generation we have that:

If n is even then at every free end of the structure we add a I-toothpick formed of two single toothpicks in vertical position.

If n is odd then at every free end of the structure we add a V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof (see the example).

a(n) gives the total number of V-toothpicks and I-toothpicks in the structure after the n-th stage.

A327333 (the first differences) gives the number of elements added at the n-th stage.

2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.

The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12-gons, concave 18-gons, concave 20-gons, and other polygons.

The structure is almost identical to the structure of A327330, but a little smaller.

The behavior seems to suggest that this sequence can be calculated with a formula, in the same way as A139250, but that is only a conjecture.

The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

It appears that A327330 shares infinitely many terms with this sequence.

Examples

			Illustration of initial terms:
.
.             /\     |/\|
.                    |  |
.
n:     0       1       2
a(n):  0       1       3
After two generations there are only one V-toothpick and two I-toothpicks in the structure, so a(2) = 1 + 2 = 3 (note that in total there are 2*a(2)= 2*3 = 6 single toothpicks of length 1).
		

Links

• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A139250 (normal toothpicks), A160164 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), A296612, A323641, A323642, A327333 (first differences), A327330 (another version).

For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.

Formula

Conjecture: a(2^k) = A327330(2^k), k >= 0.