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A296510 Toothpick sequence on triangular grid (see Comments lines for definition). 7
0, 1, 3, 7, 13, 19, 25, 31, 41, 57, 77, 93, 103, 109, 119, 135, 159, 187, 219, 247, 279, 319, 369, 409, 431, 439, 449, 465, 489, 517, 549, 581, 621, 677, 751, 827, 891, 933, 969, 1009, 1071, 1147, 1237, 1317, 1405, 1507, 1629, 1725, 1775, 1789, 1799, 1815, 1839, 1867, 1899, 1931, 1971, 2027, 2101, 2177, 2241 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

We use toothpicks of length 2, the same as the toothpick cellular automaton of A139250, but here we are on triangular grid, hence we have three axes, not two.

The Toothpicks are alternately arranged on the three axes in a rotating cycle.

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

A296511 (the first differences) gives the number of toothpicks added at n-th stage.

The structure reveals that some cellular automata that have recurrent periods can be represented by irregular triangles of first differences whose row lengths are the terms of A011782 multiplied by k (instead of powers of 2), where k is the length of their "word". In this case the word should be "abc", therefore k = 3. In the case of the cellular automaton with normal toothpicks (A139250) the word should be "ab", therefore k = 2.

For more information about the "word" of a cellular automaton see A296612.

Note that due to the unusual orientation of the polygons that are located on the edges of the structure, the image of this cellular automaton resembles the photo of an object that is rotating.

Note that between other polygons the structure contains the same "petals" as the floret pentagonal tiling.

Apparently the graph could be similar to the graph of A151907.

LINKS

Table of n, a(n) for n=0..60.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

The Poly Pages, Polyiamonds

Wikipedia, Floret pentagonal tiling

Index entries for sequences related to cellular automata

Index entries for sequences related to toothpick sequences

EXAMPLE

After 49 stages in every 60-degree wedge of the mentioned dodecagon we can see six kind of closed regions as shown below:

----------------------------------------------------------------------------------

Polygon                    Sides's length  Perimeter   Area  Quantity  Total area

----------------------------------------------------------------------------------

Triangle                   [1,1,1]             3         1     100        100

Rhombus (diamond)          [2,2,2,2]           8         8       5         40

Trapeze                    [1,2,3,2]           8         8      35        280

Irregular pentagon (petal) [1,1,1,2,2]         7         7      58        406

Irregular pentagon         [1,1,3,2,4]        11        15       1         15

Hexagon                    [1,1,1,1,1,1]       6         6      20        120

----------------------------------------------------------------------------------

Subtotal per wedge                                             219        961

.

Then we have:

Subtotal of the six wedges                                    1308       5766

Shared triangle            [1,1,1]             3         1       2          2

----------------------------------------------------------------------------------

Total of the structure after 49 stages                        1306       5764

.

CROSSREFS

Cf. A151907, A160160, A296511 (first differences), A296612.

Cf. A160120 (word "a"), A139250 (word "ab"), A299476 (word "abcb"), A299478 (word "abcbc").

Sequence in context: A310264 A144917 A102828 * A278448 A117679 A310265

Adjacent sequences:  A296507 A296508 A296509 * A296511 A296512 A296513

KEYWORD

nonn,look

AUTHOR

Omar E. Pol, Dec 14 2017

STATUS

approved

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Last modified January 21 04:53 EST 2020. Contains 331104 sequences. (Running on oeis4.)