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%I #18 Feb 24 2021 02:48:19
%S 0,2,8,16,30,52,82,104,142,196,266,288,326,380,450,504,606,756,890,
%T 912,950,1004,1074,1128,1230,1380,1514,1568,1670,1820,1986,2168,2494,
%U 2900,3162,3184,3222,3276,3346,3400,3502,3652,3786,3840,3942,4092,4258,4440
%N Q-toothpick sequence starting with two opposite Q-toothpicks centered at the same grid point.
%C The sequence gives the number of Q-toothpicks in the structure after n-th stage.
%C A187217 (the first differences) gives the number of Q-toothpicks added at n-th stage.
%C Note that in the Q-toothpick structure sometimes there is also an internal growth of Q-toothpicks.
%C For more information see A187210.
%H Nathaniel Johnston, <a href="/A187216/b187216.txt">Table of n, a(n) for n = 0..200</a>
%H Nathaniel Johnston, <a href="/A187216/a187216.c.txt">C program for computing terms</a>
%H 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 Nathaniel Johnston, <a href="http://www.nathanieljohnston.com/2011/03/the-q-toothpick-cellular-automaton/">The Q-Toothpick Cellular Automaton</a>
%H 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>
%e On the infinite square grid at stage 0 we start with no Q-toothpicks.
%e At stage 1 we place two opposite Q-toothpicks centered at (0,0). One of the Q-toothpicks lies on the first quadrant with its endpoints at (0,1) and (1,0). The other Q-toothpick lies on the third quadrant with its endpoints at (0,-1) and (-1,0). So a(1) = 2. There are 4 exposed endpoints.
%e At stage 2 we place 6 Q-toothpicks, so a(2) = 2+6 = 8.
%e At stage 3 we place 8 Q-toothpicks, so a(3) = 8+8 = 16.
%e At stage 4 we place 14 Q-toothpicks, so a(4) = 16+14 = 30.
%e After 4 stages in the Q-toothpick structure there are 1 circle, 2 "heads" and 12 exposed endpoints.
%Y Cf. A139250, A160120, A160164, A187210, A187212, A187217, A187220.
%K nonn
%O 0,2
%A _Omar E. Pol_, Mar 30 2011
%E a(15) - a(47) from _Nathaniel Johnston_, Apr 15 2011
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