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A211000
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Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks in which the inflection points are the prime numbers A000040.
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12
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0, 0, 1, 1, 2, 0, 3, -1, 4, -2, 3, -3, 2, -4, 3, -5, 4, -6, 3, -7, 2, -6, 3, -5, 4, -4, 3, -3, 2, -2, 3, -1, 4, -2, 3, -3, 2, -4, 3, -5, 4, -6, 3, -7, 2, -6, 3, -5, 4, -4, 3, -3, 2, -4, 3, -5, 4, -4, 3, -3, 2, -2, 3, -1, 4, 0, 3, 1, 2, 0, 3, -1, 4, 0
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OFFSET
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0,5
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COMMENTS
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On the infinite square grid the structure looks like a column of tangent circles of radius 1. The structure arises from the prime numbers A000040. The behavior seems to be as modular arithmetic but in a growing structure. The values on the axis "x" are easy to predict (see A211010). On the other hand the values on the axis "y" do not seem to be predictable (see A211011). This is a member of the family of the structures or curves mentioned in A210838. The odd numbers > 1 are located on the main axis of the structure. Note that here the Q-toothpicks can be superposed. For the definition of Q-toothpicks see A187210. A211021 gives the number of stage where a new circle appears in the structure. For the number of circles after n-th stage see A211020. For the location of the centers of the circles see A211022. For the sums of the visible nodes after n-th stage see A211024.
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LINKS
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Table of n, a(n) for n=0..73.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences
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EXAMPLE
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We start at stage 0 with no Q-toothpicks.
At stage 1 we place a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1).
At stage 2 we place a Q-toothpick centered at (1,0) with its endpoints at (1,1) and (2,0). Since 2 is a prime number we have that the end of the curve is also an inflection point.
At stage 3 we place a Q-toothpick centered at (3,0) with its endpoints at (2,0) and (3,-1). Since 3 is a prime number we have that the end of the curve is also an inflection point.
At stage 4 we place a Q-toothpick centered at (3,-2) with its endpoints at (3,-1) and (4,-2).
-------------------------------------
. The end as
. Pair inflection
n (x y) point
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0 0, 0, -
1 1, 1, -
2 2, 0, Yes
3 3, -1, Yes
4 4, -2, -
5 3, -3, Yes
6 2, -4, -
7 3, -5, Yes
8 4, -6, -
9 3, -7, -
10 2, -6, -
11 3, -5, Yes
...
Illustration of the nodes of the structure:
-----------------------------------------------------
After 9 stages After 10 stages After 11 stages
-----------------------------------------------------
.
. 1 1 1
. 0 2 0 2 0 2
. 3 3 3
. 4 4 4
. 5 5 5
. 6 6 6
. 7 7 11
. 8 10 8 10 8
. 9 9 9
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CROSSREFS
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Bisections: A211010, A211011.
Cf. A187210, A210838, A210841, A211001-A211003, A211020-A211024.
Sequence in context: A215589 A274185 A161162 * A025636 A025637 A195826
Adjacent sequences: A210997 A210998 A210999 * A211001 A211002 A211003
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KEYWORD
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sign
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AUTHOR
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Omar E. Pol, Mar 30 2012
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STATUS
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approved
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