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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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14
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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 the n-th stage see A211020. For the location of the centers of the circles see A211022. For the sums of the visible nodes after the n-th stage see A211024.
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LINKS
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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).
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. 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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MATHEMATICA
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A211000[nmax_]:=Module[{walk={{0, 0}}, angle=3/4Pi, turn=Pi/2}, Do[If[!PrimeQ[n], If[n>5&&PrimeQ[n-1], turn*=-1]; angle-=turn]; AppendTo[walk, AngleVector[Last[walk], {Sqrt[2], angle}]], {n, 0, nmax-1}]; walk];
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PROG
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(PARI)
A211000(nmax) = my(walk=vector(nmax+1), turn=1, p1, p2); walk[1]=[0, 0]; if(nmax==0, return(walk)); walk[2]=[1, 1]; for(n=1, nmax-1, p1=walk[n]; p2=walk[n+1]; if(isprime(n), walk[n+2]=[2*p2[1]-p1[1], 2*p2[2]-p1[2]], if(n>5 && isprime(n-1), turn*=-1); walk[n+2]=[p2[1]-turn*(p1[2]-p2[2]), p2[2]+turn*(p1[1]-p2[1])])); walk;
(Python)
from sympy import isprime
walk, turn = [(0, 0), (1, 1)], 1
for n in range(1, nmax):
p1, p2 = walk[-2], walk[-1]
if isprime(n): # Go straight
walk.append((2*p2[0]-p1[0], 2*p2[1]-p1[1]))
else: # Turn
if n>5 and isprime(n-1): turn *= -1
walk.append((p2[0]-turn*(p1[1]-p2[1]), p2[1]+turn*(p1[0]-p2[0])))
return walk[:nmax+1]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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