

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 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

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 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

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



