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A327332
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"Concave pentagon" toothpick sequence, starting with a V-toothpick (see Comments for precise definition).
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4
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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
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OFFSET
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0,3
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COMMENTS
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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:
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* *
* * * *
* * * *
* * *
* 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.
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LINKS
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FORMULA
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Conjecture: a(2^k) = A327330(2^k), k >= 0.
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EXAMPLE
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Illustration of initial terms:
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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).
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CROSSREFS
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For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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