A296510 Toothpick sequence on triangular grid (see Comments lines for definition).

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

Offset: 0

Author

Omar E. Pol, Dec 14 2017

Keywords

• nonn
• look

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.

Examples

			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
		

Links

• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• The Poly Pages, Polyiamonds
• Rémy Sigrist, Illustration of the construction at generation 3*256
• Wikipedia, Floret pentagonal tiling
• Index entries for sequences related to cellular automata
• Index entries for sequences related to toothpick sequences

Crossrefs

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

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