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A187210 Q-toothpick sequence (see Comments for precise definition).

%I A187210 #158 Nov 22 2022 22:14:22
%S A187210 0,1,5,12,24,46,66,88,128,182,222,244,284,338,394,464,584,718,790,812,
%T A187210 852,906,962,1032,1152,1286,1374,1444,1564,1714,1882,2128,2488,2814,
%U A187210 2950,2972,3012,3066,3122,3192,3312,3446,3534,3604,3724,3874,4042,4288,4648,4974,5126,5196,5316,5466,5634,5880,6240,6582,6814,7060,7436,7890,8458,9296,10328
%N A187210 Q-toothpick sequence (see Comments for precise definition).
%C A187210 We define a "Q-toothpick" to be a quarter-circle. The length of a Q-toothpick is equal to Pi/2 = 1.570796...
%C A187210 In order to construct this sequence we use the following rules:
%C A187210 - Each new Q-toothpick must lie on the square grid (or circular grid) such that the Q-toothpick endpoints coincide with two opposite vertices of a unit square.
%C A187210 - Each exposed endpoint of the Q-toothpicks of the old generation must be touched by the endpoints of two q-toothpicks of new generation without creating a corner or vertex between these three arcs such that the couple of new Q-toothpicks should look like a "gullwing".
%C A187210 Note that in the Q-toothpick structure sometimes there is also an internal growth of the Q-toothpicks.
%C A187210 The sequence gives the number of Q-toothpicks in the structure after n stages. A187211 (the first differences) gives the number of Q-toothpicks added at n-th stage.
%C A187210 Note that the structure of the Q-toothpick cellular automaton contains distinct types of geometrical figures, for example: circles, diamonds, hearts, heads or flower vases (which appears only on the main diagonal) and also an infinity family of objects (blobs) where every object is a closed region which contains 2^k virtual circles with radius 1 and 2^k-1 virtual diamonds, for example: a 2 X 2 object is a closed region which contains exactly four virtual circles and three virtual diamonds, a 2 X 4 object is a closed region which contains exactly 8 virtual circles and 7 virtual diamonds, etc. Note that a "heart" can be considered a 1 X 2 object which contains two virtual circles and a virtual diamond. What is the better name for these figures? Note that there is a correspondence between this last family of objects and the squares and rectangles of the hidden crosses in the toothpick structure of A139250. For more information about the connection with the toothpick sequence see A139250, A160164 and A187220.
%C A187210 It appears that the number of hearts present in the n-th generation equals the number of rectangles of area = 2 present in the (n-2)nd generation of the toothpick structure of A139250, assuming the toothpicks have length 2, if n >= 3 (see also A188346 and A211008). - _Omar E. Pol_, Sep 30 2012
%C A187210 From _Omar E. Pol_, Jan 23 2016: (Start)
%C A187210 Consider the initial Q-toothpick with the virtual center at (0,0) and its endpoints at (0,1) and (1,0).
%C A187210 If n is a power of 2 plus 2 and n >> 1 then the structure of this C.A. essentially looks like a square which contains four parts (or sectors) as follows:
%C A187210 1) NW quadrant, but whose origin is at (-1,1). In this quadrant the number of Q-toothpicks after n generations equals the number of toothpicks in the toothpick structure of A139250 after n-2 generations, if n >= 2. Note that here the toothpick sequence A139250 is represented with Q-toothpicks arranged in an asymmetric structure.
%C A187210 2) SE quadrant, but whose origin is at (1,-1). This quadrant is a reflected copy of the NW quadrant, hence the number of Q-toothpicks after n generations equals A139250(n-2), n >= 2, the same as in the NW quadrant.
%C A187210 3) SW quadrant, but with the origin in the first quadrant at (1,1). In this quadrant the number of Q-toothpicks after n generations is 1 + A267694(n-1), n >= 1.
%C A187210 4) NE quasi-quadrant. In this sector the number of Q-toothpicks after n-generations is A267698(n-2) - 2, if n >= 6. (End)
%C A187210 After the first few generations the behavior is similar to the Gullwing cellular automaton of A187220, but the growth is faster than A187220 and thus it's much faster than A139250. For an animation see Applegate's The movie version in the Links section. - _Omar E. Pol_, Sep 13 2016
%D A187210 A. Adamatzky and G. J. Martinez, Designing Beauty: The Art of Cellular Automata, Springer, 2016, pages 59, 62 (note that the Q-toothpick cellular automaton is erroneously attributed to _Nathaniel Johnston_).
%H A187210 Nathaniel Johnston, <a href="/A187210/b187210.txt">Table of n, a(n) for n = 0..177</a>
%H A187210 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A187210 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
%H A187210 Elisabet Edvardsson and Eva Mossberg, <a href="https://www.oldcitypublishing.com/journals/jca-home/jca-issue-contents/jca-volume-14-number-1-2-2019/jca-14-1-2-p-51-68/">The Q-toothpick Cellular Automaton</a>, Journal of Cellular Automata, Vol. 14, Issue 1-2, (2019), p. 51-68.
%H A187210 Nathaniel Johnston, <a href="/A187210/a187210.gif">Animation of first 19 generations</a>
%H A187210 Nathaniel Johnston, <a href="http://www.jstor.org/stable/10.4169/college.math.j.46.1.fm">Illustration of a(5) = 46</a>, "Front Matter" 2015. The College Mathematics Journal 46 (1). Mathematical Association of America: 1-1. doi:10.4169/college.math.j.46.1.fm.
%H A187210 Nathaniel Johnston, <a href="http://www.nathanieljohnston.com/2011/03/the-q-toothpick-cellular-automaton/">The Q-Toothpick Cellular Automaton</a>
%H A187210 Nathaniel Johnston, <a href="http://www.conwaylife.com/forums/viewtopic.php?f=11&amp;t=663">The Q-Toothpick post in ConwayLife.com</a>
%H A187210 Omar E. Pol, <a href="/A187210/a187210.jpg">Illustration of initial terms</a>
%H A187210 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 A187210 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A187210 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F A187210 a(0)=0; a(1)=1; a(n) = 2*A139250(n-2) + A267698(n-2) + A267694(n-1) + m, where m = 3 if 2 <= n <= 5 and m = -1 if n>=6 (note that 2*A139250(n-2) can be replaced with A160164(n-2)). - _Omar E. Pol_, Jan 23 2016
%F A187210 a(n) = A187220(n-1) + A267698(n-2) + A267694(n-1) + m, where m = 2 if 2 <= n <= 5 and m = -2 if n >= 6. - _Omar E. Pol_, Sep 13 2016
%e A187210 From _Omar E. Pol_, Apr 02 2016: (Start)
%e A187210 Examples that are related to the toothpick sequence A139250 (see the first formula):
%e A187210 For n = 5 we have that A139250(5-2) = 7, A267698(5-2) = 13, A267694(5-1) = 16 and m = 3, so a(5) = 2*7 + 13 + 16 + 3 = 46.
%e A187210 For n = 6 we have that A139250(6-2) = 11, A267698(6-2) = 25, A267694(6-1) = 20 and m = -1, so a(6) = 2*11 + 25 + 20 - 1 = 66. (End)
%e A187210 From _Omar E. Pol_, Sep 13 2016: (Start)
%e A187210 Examples that are related to the Gullwing sequence A187220 (see the second formula):
%e A187210 For n = 5 we have that A187220(5-1) = 15, A267698(5-2) = 13, A267694(5-1) = 16 and m = 2, so a(5) = 15 + 13 + 16 + 2 = 46.
%e A187210 For n = 6 we have that A187220(6-1) = 23, A267698(6-2) = 25, A267694(6-1) = 20 and m = -2, so a(6) = 23 + 25 + 20 - 2 = 66. (End)
%Y A187210 Cf. A139250, A160164, A187211, A187212, A187220, A188344, A188345, A188346, A267694, A267698.
%K A187210 nonn
%O A187210 0,3
%A A187210 _Omar E. Pol_, Mar 07 2011
%E A187210 Terms a(8) and beyond from _Nathaniel Johnston_, Mar 26 2011
%E A187210 Comments edited by _Omar E. Pol_, Mar 28 2011
%E A187210 Second rule clarified by _Omar E. Pol_, Apr 06 2011