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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks (A160164) and V-toothpicks (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 I-toothpick 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 V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof.

If n is odd then we add I-toothpicks in vertical position (see the example).

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

A327331 (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 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 V-toothpick, 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 I-toothpicks and two V-toothpicks 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 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), 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

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Last modified June 6 01:16 EDT 2020. Contains 334858 sequences. (Running on oeis4.)