This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A296510 #42 Nov 22 2022 11:57:46 %S A296510 0,1,3,7,13,19,25,31,41,57,77,93,103,109,119,135,159,187,219,247,279, %T A296510 319,369,409,431,439,449,465,489,517,549,581,621,677,751,827,891,933, %U A296510 969,1009,1071,1147,1237,1317,1405,1507,1629,1725,1775,1789,1799,1815,1839,1867,1899,1931,1971,2027,2101,2177,2241 %N A296510 Toothpick sequence on triangular grid (see Comments lines for definition). %C A296510 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. %C A296510 The Toothpicks are alternately arranged on the three axes in a rotating cycle. %C A296510 a(n) gives the number of toothpicks in the structure after n-th stage. %C A296510 A296511 (the first differences) gives the number of toothpicks added at n-th stage. %C A296510 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. %C A296510 For more information about the "word" of a cellular automaton see A296612. %C A296510 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. %C A296510 Note that between other polygons the structure contains the same "petals" as the floret pentagonal tiling. %C A296510 Apparently the graph could be similar to the graph of A151907. %H A296510 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a> %H A296510 The Poly Pages, <a href="http://www.recmath.org/PolyPages/PolyPages/index.htm?Polyiamonds.htm">Polyiamonds</a> %H A296510 Rémy Sigrist, <a href="/A296511/a296511.png">Illustration of the construction at generation 3*256</a> %H A296510 Wikipedia, <a href="https://en.wikipedia.org/wiki/Snub_trihexagonal_tiling#Floret_pentagonal_tiling">Floret pentagonal tiling</a> %H A296510 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a> %H A296510 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a> %e A296510 After 49 stages in every 60-degree wedge of the mentioned dodecagon we can see six kind of closed regions as shown below: %e A296510 ---------------------------------------------------------------------------------- %e A296510 Polygon Sides's length Perimeter Area Quantity Total area %e A296510 ---------------------------------------------------------------------------------- %e A296510 Triangle [1,1,1] 3 1 100 100 %e A296510 Rhombus (diamond) [2,2,2,2] 8 8 5 40 %e A296510 Trapeze [1,2,3,2] 8 8 35 280 %e A296510 Irregular pentagon (petal) [1,1,1,2,2] 7 7 58 406 %e A296510 Irregular pentagon [1,1,3,2,4] 11 15 1 15 %e A296510 Hexagon [1,1,1,1,1,1] 6 6 20 120 %e A296510 ---------------------------------------------------------------------------------- %e A296510 Subtotal per wedge 219 961 %e A296510 . %e A296510 Then we have: %e A296510 Subtotal of the six wedges 1308 5766 %e A296510 Shared triangle [1,1,1] 3 1 2 2 %e A296510 ---------------------------------------------------------------------------------- %e A296510 Total of the structure after 49 stages 1306 5764 %Y A296510 Cf. A151907, A160160, A296511 (first differences), A296612. %Y A296510 Cf. A160120 (word "a"), A139250 (word "ab"), A299476 (word "abcb"), A299478 (word "abcbc"). %K A296510 nonn,look %O A296510 0,3 %A A296510 _Omar E. Pol_, Dec 14 2017