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 A220494 #22 Nov 13 2018 03:11:39 %S A220494 0,1,3,7,11,15,19,27,35,39,43,51,59,67,75,91,107,111,115,123,131,139, %T A220494 147,163,179,187,195,211,227,243,259,291,323,327,331,339,347,355,363, %U A220494 379,395,403,411,427,443,459,475,507,539,547,555,571,587,603 %N A220494 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "wide" triangle of the first kind. %C A220494 This cellular automaton uses toothpicks of length 1 and D-toothpicks of length 2^(1/2). Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints. %C A220494 On the semi-infinite square grid we start with no elements, so a(0) = 0. At stage 1, we place a single toothpick in vertical direction at (0,0),(0,1), so a(1) = 1. Note that there is only one exposed toothpick endpoint. %C A220494 The rules for adding new elements are as follows. Each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees. The endpoints of the toothpicks of the old generation that are perpendiculars to the initial toothpick remain exposed forever. Overlapping is prohibited. %C A220494 The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A220495 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage. %C A220494 It appears that the structure has fractal behavior related to powers of 2. It appears that this cellular automaton has a surprising connection with the Sierpinski triangle, but here the structure is more complex. %C A220494 For a similar version see A220496. For other more complex versions see A194440, A220520. First differs from A194440 (and from A220520) at a(12). %H A220494 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp056.jpg">Illustration of the structure after 32 stages</a> %H A220494 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 A220494 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a> %Y A220494 Cf. A047999, A139250, A194270, A194440, A220495, A220496, A220520. %K A220494 nonn %O A220494 0,3 %A A220494 _Omar E. Pol_, Dec 23 2012