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%I #80 Feb 24 2021 02:48:18
%S 0,2,6,14,22,30,46,70,86,94,110,134,158,190,246,310,342,350,366,390,
%T 414,446,502,566,606,638,694,766,846,966,1142,1302,1366,1374,1390,
%U 1414,1438,1470,1526,1590,1630,1662,1718,1790
%N Number of toothpicks after n-th stage in the I-toothpick structure of A139250.
%C From _Omar E. Pol_, Mar 12 2011, Mar 15 2011, Mar 22 2011, Mar 25 2011: (Start)
%C We define an "I-toothpick" to consist of two connected toothpicks, as a bar of length 2. An I-toothpick with length 2 is formed by two toothpicks with length 1.
%C Note that in the physical model of the toothpick structure of A139250 the midpoint of a wooden toothpick of the new generation is superimposed on the endpoint of a wooden toothpick of the old generation. However, in the physical model of the I-toothpick structure the wooden toothpicks are not overlapping because all wooden toothpicks are connected by their endpoints.
%C a(n) is also the number of components after n-th stage in the toothpick structure of A139250, assuming the toothpicks have length 2.
%C Also, gullwing sequence starting from two opposite "gulls" (as a reflected gull in flight) such that the distance between their midpoints is equal to 2 (See A187220). The sequence gives the number of gulls in the structure after n-th stage.
%C Note that there is a correspondence between the gullwing structure and the I-toothpick structure, for example: a pair of opposite gulls in horizontal position in the gullwing structure is equivalent to a vertical I-toothpick with length 4 in the I-toothpick structure, such that the midpoint of each horizontal gull coincides with the midpoint of each vertical toothpick of the I-toothpick.
%C It appears this is also the connection between A147562 (the Ulam-Warburton cellular automaton) and the toothpick sequence A139250. The behavior of the function is similar to A147562 but here the structure is more complex. See Plot 2 button: A147562 vs A160164. See also A147562 vs A187220.
%C Also, B-toothpick sequence starting from two opposite "bells" such that the distance between their midpoints is equal to 4 (See A187220). We define a "B-toothpick" to consist of four arcs of length Pi/2 forming a "bell" similar to the Gauss function. A bell-shaped toothpick or B-toothpick or simply "bell" is formed by four Q-toothpicks (see A187210). A B-toothpick has length 2*Pi. The sequence gives the number of bells in the structure after n stages.
%C We can see a correspondence between this structure and the I-toothpick structure of A139250. In this case, for example, a pair of opposite bells in horizontal position is equivalent to a vertical I-toothpick with length 8 in the I-toothpick structure, such that the midpoint of each horizontal bell coincides with the midpoint of each vertical toothpick of the I-toothpick.
%C Also, there is a fourth structure formed by isosceles right triangles, starting from two opposite triangles, since gulls or bells can be replaced by this type of triangles.
%C Note that the size of the toothpicks, gulls, bells and isosceles right triangles can be adjusted such that two or more of these structures can be overlaid.
%C (End)
%C The graph of this sequence is very close to the graphs of both A147562 and A169707 (see Plot 2). - _Omar E. Pol_, Feb 16 2015
%C It appears that a(n) is also the total number of ON cells after n-th stage in the half structure of the cellular automaton described in A169707 plus the total number of ON cells after n+1 stages in the half structure of the mentioned cellular automaton, without its central cell. See the illustration of the NW-NE-SE-SW version in A169707. - _Omar E. Pol_, Jul 26 2015
%C On the infinite Cairo pentagonal tiling consider the symmetric figure formed by two non-adjacent pentagons connected by a line segment joining two trivalent nodes. At stage 1 we start with one of these figures turned ON. a(n) is the number of ON cells in the structure after n-th stage, so a(1) = 2. The rule for the next stages is that the concave part of the figures of the new generation must be adjacent to the complementary convex part of the figures of the old generation. - _Omar E. Pol_, Mar 29 2018
%H N. J. A. Sloane, <a href="/A160164/b160164.txt">Table of n, a(n) for n = 0..16384</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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling">Cairo pentagonal tiling</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F a(n) = 2*A139250(n).
%F a(n) = A187220(n+1) - 1. - _Omar E. Pol_, Mar 12 2011, Mar 22 2011
%F It appears that a(n) = A169707(n) + A170903(n), n >= 1. - _Omar E. Pol_, Feb 15 2015
%F It appears that a(n) = (A169707(n) - 1)/2 + (A169707(n+1) - 1)/2, n >= 1. - _Omar E. Pol_, Jul 24 2015
%e From _Omar E. Pol_, Aug 12 2013: (Start)
%e Illustration of initial terms:
%e . _ _ _ _
%e . _ _ _ _ |_ _ _ _| |_ _ _ _|
%e . _ _ |_ _| |_ _| | |_ _| | _|_|_ _|_|_
%e . | | | | | | | | | | | | | |
%e . | _|_ |_|_| |_|_| |_|_| _ _|_|_|_ _
%e . | | _|_ _|_ |_|_ _|_| |_|_ _|_|
%e . | | _|_ _|_
%e .
%e . 2 6 14 22 30 46
%e .
%e (End)
%t CoefficientList[Series[(2 x / ((1 - x) (1 + 2 x))) (1 + 2 x Product[1 + x^(2^k - 1) + 2 x^(2^k), {k, 0, 20}]), {x, 0, 53}], x] (* _Vincenzo Librandi_, Feb 15 2015 *)
%Y Cf. A139250, A147562, A169707, A170903, A187210, A187220.
%K nonn
%O 0,2
%A _Omar E. Pol_, Jun 01 2009
%E Zero inserted, more terms and edited by _Omar E. Pol_, Mar 12 2011
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