

A296510


Toothpick sequence on triangular grid (see Comments lines for definition).


8



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
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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 nth stage.
A296511 (the first differences) gives the number of toothpicks added at nth 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



EXAMPLE

After 49 stages in every 60degree 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



KEYWORD



AUTHOR



STATUS

approved



