

A160164


Number of toothpicks after nth stage in the Itoothpick structure of A139250.


19



0, 2, 6, 14, 22, 30, 46, 70, 86, 94, 110, 134, 158, 190, 246, 310, 342, 350, 366, 390, 414, 446, 502, 566, 606, 638, 694, 766, 846, 966, 1142, 1302, 1366, 1374, 1390, 1414, 1438, 1470, 1526, 1590, 1630, 1662, 1718, 1790
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OFFSET

0,2


COMMENTS

From Omar E. Pol, Mar 12 2011, Mar 15 2011, Mar 22 2011, Mar 25 2011: (Start)
We define an "Itoothpick" to consist of two connected toothpicks, as a bar of length 2. An Itoothpick with length 2 is formed by two toothpicks with length 1.
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 Itoothpick structure the wooden toothpicks are not overlapping because all wooden toothpicks are connected by their endpoints.
a(n) is also the number of components after nth stage in the toothpick structure of A139250, assuming the toothpicks have length 2.
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 nth stage.
Note that there is a correspondence between the gullwing structure and the Itoothpick structure, for example: a pair of opposite gulls in horizontal position in the gullwing structure is equivalent to a vertical Itoothpick with length 4 in the Itoothpick structure, such that the midpoint of each horizontal gull coincides with the midpoint of each vertical toothpick of the Itoothpick.
It appears this is also the connection between A147562 (the UlamWarburton 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.
Also, Btoothpick sequence starting from two opposite "bells" such that the distance between their midpoints is equal to 4 (See A187220). We define a "Btoothpick" to consist of four arcs of length Pi/2 forming a "bell" similar to the Gauss function. A bellshaped toothpick or Btoothpick or simply "bell" is formed by four Qtoothpicks (see A187210). A Btoothpick has length 2*Pi. The sequence gives the number of bells in the structure after n stages.
We can see a correspondence between this structure and the Itoothpick structure of A139250. In this case, for example, a pair of opposite bells in horizontal position is equivalent to a vertical Itoothpick with length 8 in the Itoothpick structure, such that the midpoint of each horizontal bell coincides with the midpoint of each vertical toothpick of the Itoothpick.
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.
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.
(End)
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
It appears that a(n) is also the total number of ON cells after nth 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 NWNESESW version in A169707.  Omar E. Pol, Jul 26 2015
On the infinite Cairo pentagonal tiling consider the symmetric figure formed by two nonadjacent 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 nth 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


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..16384
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.],
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Wikipedia, Cairo pentagonal tiling
Index entries for sequences related to cellular automata


FORMULA

a(n) = 2*A139250(n).
a(n) = A187220(n+1)  1.  Omar E. Pol, Mar 12 2011, Mar 22 2011
It appears that a(n) = A169707(n) + A170903(n), n >= 1.  Omar E. Pol, Feb 15 2015
It appears that a(n) = (A169707(n)  1)/2 + (A169707(n+1)  1)/2, n >= 1.  Omar E. Pol, Jul 24 2015


EXAMPLE

From Omar E. Pol, Aug 12 2013: (Start)
Illustration of initial terms:
. _ _ _ _
. _ _ _ _ _ _ _ _ _ _ _ _
. _ _ _ _ _ _  _ _  ___ ___
.              
.  __ __ __ __ _ ____ _
.   __ __ __ __ __ __
.   __ __
.
. 2 6 14 22 30 46
.
(End)


MATHEMATICA

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 *)


CROSSREFS

Cf. A139250, A147562, A169707, A170903, A187210, A187220.
Sequence in context: A268641 A162796 A172304 * A074729 A099901 A119846
Adjacent sequences: A160161 A160162 A160163 * A160165 A160166 A160167


KEYWORD

nonn


AUTHOR

Omar E. Pol, Jun 01 2009


EXTENSIONS

Zero inserted, more terms and edited by Omar E. Pol, Mar 12 2011


STATUS

approved



