

A327330


"Concave pentagon" toothpick sequence (see Comments for precise definition).


3



0, 1, 3, 7, 11, 15, 23, 33, 41, 45, 53, 63, 75, 89, 111, 133, 149, 153, 161, 171, 183, 197, 219, 241, 261, 275, 299, 327, 361, 403, 463, 511, 547, 551, 559, 569, 581, 595, 617, 639, 659, 673, 697, 725, 759, 801, 861, 909, 949, 967, 995, 1029, 1075, 1125, 1183, 1233, 1281, 1321, 1389, 1465, 1549, 1657
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OFFSET

0,3


COMMENTS

This arises from a hybrid cellular automaton on a triangular grid formed of Itoothpicks (A160164) and Vtoothpicks (A161206).
The surprising fact is that after 2^k stages the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two quadrilaterals (Q1 and Q2), both with their largest sides in vertical position, as shown below:
.
* *
* * * *
* * * *
* * *
* Q1 * Q2 *
* * * *
* * * *
* * * *
* * * *
* * E * *
* * * *
* * * *
** **
* * * * * * * * * *
.
Note that for n >> 1 both quadrilaterals look like right triangles.
Every polygon 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 Itoothpick formed of two single toothpicks in vertical position, 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 Vtoothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof.
If n is odd then we add Itoothpicks in vertical position (see the example).
a(n) gives the total number of Itoothpicks and Vtoothpicks in the structure after the nth stage.
A327331 (the first differences) gives the number of elements added at the nth stage.
2*a(n) gives the total number of single toothpicks of length 1 after the nth stage.
The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12gons, concave 18gons, concave 20gons, and other polygons.
The structure is almost identical to the structure of A327332, but a little larger at the upper edge.
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.
For another version, very similar, starting with a Vtoothpick, see A327332, which it appears that shares infinitely many terms with this sequence.


LINKS

Table of n, a(n) for n=0..61.
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


FORMULA

Conjecture: a(2^k) = A327332(2^k), k >= 0.


EXAMPLE

Illustration of initial terms:
.
 /\ /\
    
/ \ / \
 
n : 0 1 2 3
a(n): 0 1 3 7
After three generations there are five Itoothpicks and two Vtoothpicks in the structure, so a(3) = 5 + 2 = 7 (note that in total there are 2*a(3) = 2*7 = 14 single toothpicks of length 1).


CROSSREFS

First differs from A231348 at a(11).
Cf. A047999, A139250 (normal toothpicks), A160164 (Itoothpicks), A160722 (a concave pentagon with triangular cells), A161206 (Vtoothpicks), A296612, A323641, A323642, A327331 (first differences), A327332 (another version).
For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.
Sequence in context: A172306 A309274 A112714 * A231348 A194444 A220524
Adjacent sequences: A327327 A327328 A327329 * A327331 A327332 A327333


KEYWORD

nonn


AUTHOR

Omar E. Pol, Sep 01 2019


STATUS

approved



