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 A323646 #46 Dec 07 2019 00:45:03 %S A323646 0,1,3,5,9,15,21,27,39,53,65,71,83,97,113,131,163,197,217,223,235,249, %T A323646 265,283,315,349,373,391,423,461,505,567,659,741,777,783,795,809,825, %U A323646 843,875,909,933,951,983,1021,1065,1127,1219,1301,1341,1359,1391,1429,1473,1535,1627,1713,1773,1835,1931 %N A323646 "Letter A" toothpick sequence (see Comments for precise definition). %C A323646 This arises from a hybrid cellular automaton formed of toothpicks of length 2 and D-toothpicks of length 2*sqrt(2). %C A323646 For the construction of the sequence the rules are as follows: %C A323646 On the infinite square grid at stage 0 there are no toothpicks, so a(0) = 0. %C A323646 For the next n generations we have that: %C A323646 At stage 1 we place a toothpick of length 2 in the horizontal direction, centered at [0,0], so a(1) = 1. %C A323646 If n is even we add D-toothpicks. Each new D-toothpick must have its midpoint touching the endpoint of exactly one existing toothpick. %C A323646 If the x-coordinate of the middle point of the D-toothpick is negative then the D-toothpick must be placed in the NE-SW direction. %C A323646 If the x-coordinate of the middle point of the D-toothpick is positive then the D-toothpick must be placed in the NW-SE direction. %C A323646 If n is odd we add toothpicks in horizontal direction. Each new toothpick must have its midpoint touching the endpoint of exactly one existing D-toothpick. %C A323646 The sequence gives the number of toothpicks and D-toothpicks after n stages. %C A323646 A323647 (the first differences) gives the number of elements added at the n-th stage. %C A323646 Note that if n >> 1 at the end of every cycle the structure looks like a "volcano", or in other words, the structure looks like a trapeze which is almost an isosceles right triangle. %C A323646 The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612. %H A323646 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 A323646 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a> %H A323646 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a> %F A323646 a(n) = 1 + A160730(n-1), n >= 1. %F A323646 a(n) = 1 + 2*A168112(n-1), n >= 1. %e A323646 After two generations the structure looks like a letter "A" which is formed by a initial I-toothpick (or a toothpick of length 2), placed in horizontal direction, and two D-toothpicks each of length 2*sqrt(2) as shown below, so a(2) = 3. %e A323646 Note that angle between both D-toothpicks is 90 degrees. %e A323646 . %e A323646 * %e A323646 * * %e A323646 * * * * * %e A323646 * * %e A323646 * * %e A323646 . %e A323646 After three generations the structure contains three horizontal toothpicks and two D-toothpicks as shown below, so a(3) = 5. %e A323646 . %e A323646 * %e A323646 * * %e A323646 * * * * * %e A323646 * * %e A323646 * * * * * * * * * * %e A323646 . %Y A323646 Cf. A139250, A168112, A160730, A296612, A323647. %Y A323646 For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A292612, A299770, A323650, A327330, A327332. %K A323646 nonn %O A323646 0,3 %A A323646 _Omar E. Pol_, Mar 07 2019