A211001 -id:A211001 - cp's OEIS frontend

A211000 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.

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

Offset: 0

Omar E. Pol, Mar 30 2012

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.

			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
-------------------------------
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
.

Paolo Xausa, Table of n, a(n) for n = 0..9999
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Bisections: A211010, A211011.

Cf. A187210, A210838, A210841, A211001-A211003, A211020-A211024, A355478, A357434.

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];
A211000[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Aug 23 2022 *)

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;
A211000(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Sep 22 2022

from sympy import isprime
def A211000(nmax):
    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]
print(A211000(100)) # Generates 101 coordinate pairs - Paolo Xausa, Sep 22 2022

A211020 Number of circles in the structure of A211000 after n-th stage.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Omar E. Pol, Mar 30 2012

nonn

For n >= 13 the structure looks like essentially a column of tangent circles of radius 1. The circles are centered on a straight line which is parallel to the axis "y". The structure arises from the prime numbers A000040.

			From _Paolo Xausa_, Jan 09 2023: (Start)
In the following diagrams the A211000 structure is shown at the end of the n-th stage (Q-toothpicks are depicted as straight lines instead of circle arcs; circles are depicted as rhombi).
n       0       5      11      13      15      34      41      65
a(n)    0       0       1       2       3       4       5       6
.
                                                                /\
                                                                \/
                                                                 \
                                                                 /
                                                        /\      /\
                                                        \/      \/
              /\      /\      /\      /\      /\/\    /\/\    /\/\
                \       \       \       \       \/      \/      \/
                 \       \       \      /\      /\      /\      /\
                 /       /       /      \/      \/      \/      \/
                        /       /\      /\      /\      /\      /\
                        \       \/      \/      \/      \/      \/
                        /\      /\      /\      /\      /\      /\
                        \/      \/      \/      \/      \/      \/
(End)

Paolo Xausa, Table of n, a(n) for n = 0..9999
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A187210, A210838, A210841.
Cf. A211000, A211001, A211002, A211003.
Cf. A211010, A211011, A211021, A211022, A211023.
Cf. A355478, A357434.

A211020[nmax_]:=Module[{ep={{0, 0}}, angle=3/4Pi, turn=Pi/2, cells}, Join[{0}, Table[If[!PrimeQ[n], If[n>5&&PrimeQ[n-1], turn*=-1]; angle-=turn]; AppendTo[ep, AngleVector[Last[ep], {Sqrt[2], angle}]]; cells=FindCycle[Graph[MapApply[UndirectedEdge, Partition[ep, 2, 1]]], {4}, All]; CountDistinct[Map[Sort, Map[First, cells, {2}]]], {n, 0, nmax-1}]]];
A211020[100] (* Paolo Xausa, Jan 06 2023 *)

Offset changed to 0 and a(0) prepended by Paolo Xausa, Jan 06 2023

A211011 Value on the axis "y" of the endpoint of the structure (or curve) of A211000 at n-th stage.

Original entry on oeis.org

0, 1, 0, -1, -2, -3, -4, -5, -6, -7, -6, -5, -4, -3, -2, -1, -2, -3, -4, -5, -6, -7, -6, -5, -4, -3, -4, -5, -4, -3, -2, -1, 0, 1, 0, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 0, 1, 0, -1, 0, 1, 2, 3, 2, 1, 2, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 9, 8, 7, 6, 5

Offset: 0

Omar E. Pol, Mar 30 2012

For n >= 13 the structure of A211000 looks like essentially 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. Note that all odd numbers > 1 are located on the main axis of the structure. For the number of circles after n-th stage see A211020. For the values on the axis "x" see A211010. For the values for the n-th prime see A211023.

			Consider the illustration of the structure of A211000:
------------------------------------------------------
.           After           After            After
.  y      9 stages        10 stages        11 stages
------------------------------------------------------
.  2
.  1        1               1                1
.  0      0   2           0   2            0   2
. -1            3               3                3
. -2              4               4                4
. -3            5               5                5
. -4          6               6                6
. -5            7               7               11
. -6              8          10   8           10   8
. -7            9               9                9
. -8
We can see that a(7) = a(11) = -5.

Paolo Xausa, Table of n, a(n) for n = 0..9999
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Bisection of A211000.

Cf. A187210, A210838, A210841, A211001-A211003, A211010, A211020-A211024.

A211011[nmax_]:=Module[{ep={0,0},angle=3/4Pi,turn=Pi/2},Join[{0},Table[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];Last[ep=AngleVector[ep,{Sqrt[2],angle}]],{n,0,nmax-1}]]];
A211011[100] (* Paolo Xausa, Jan 14 2023 *)

abs(a(n)-a(n+1)) = 1.

A211010 Value on the axis "x" of the endpoint of the structure of A211000 at n-th stage.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3

Offset: 0

Omar E. Pol, Mar 30 2012

For the values on the axis "y" see A211011.

Also [0, 1] together with the absolute values of A131731.

Paolo Xausa, Table of n, a(n) for n = 0..9999
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Bisection of A211000.
Cf. A211011 (the y's in A211000).
Cf. A131731, A164358, A210838, A210841.
Cf. A211001, A211002, A211003, A211020, A211021, A211022.

A211010[nmax_]:=PadRight[{0,1,2,3},nmax+1,{4,3,2,3}];
A211010[100] (* Paolo Xausa, Jan 06 2023 *)

From Colin Barker, Sep 15 2013: (Start)
a(n) = (6+(-i)^n+i^n)/2 where n > 1, i=sqrt(-1).
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 4.
G.f.: -x*(x+1)*(2*x^2+1) / ((x-1)*(x^2+1)). (End)

A211003 Primes congruent to 3 in the structure (or curve) of A211000.

Original entry on oeis.org

3, 31, 43, 47, 151, 179, 251, 347, 359, 367, 6571, 6599, 6607, 6619, 6659, 6679

Offset: 1

Omar E. Pol, Mar 30 2012

The behavior seems to be as modular arithmetic but in a growing structure. For n >= 13 the structure is essentially a column of tangent circles of radius 1.

Primes in A211002.

Cf. A211000, A211001, A211010, A211011, A211020-211023.

A211023 Value on the axis "y" of the endpoint of the structure of A211000 if the index is prime.

Original entry on oeis.org

0, -1, -3, -5, -5, -3, -3, -5, -5, -3, -1, 1, 1, -1, -1, 1, 3, 5, 7, 7, 5, 3, 3, 5, 5, 5, 7, 7, 5, 5, 7, 7, 5, 3, 1, -1, -3, -5, -5, -3, -1, 1, 3, 5, 5, 3, 3, 3, 3, 5, 5, 3, 1, -1, -3, -5, -7, -9, -11, -11, -9, -7, -5, -5, -7, -7, -5, -3, -1, 1, 1, -1, -1, 1, 3

Offset: 1

Omar E. Pol, Mar 31 2012

sign

a(n) is also the value on the axis "y" of the n-th inflection point in the structure of A211000.

The behavior seems to be as modular arithmetic but in a growing structure. The structure of A211000 looks like essentially a column of tangent circles of radius 1. The structure of A211000 arises from the prime numbers A000040.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A187210, A211000, A211001, A211002, A211003, A211010, A211011, A211020, A211021, A211022, A211024.

A211023[upto_]:=Module[{ep={0, 0}, angle=3/4Pi, turn=Pi/2}, Table[If[!PrimeQ[n], If[n>5&&PrimeQ[n-1], turn*=-1]; angle-=turn]; ep=AngleVector[ep, {Sqrt[2], angle}];If[PrimeQ[n+1], Last[ep], Nothing], {n, 0,upto-1}]];
A211023[500] (* Paolo Xausa, Jan 14 2023 *)

a(n) = A211011(A000040(n)).

A211002 Numbers congruent to 3 in the structure (or curve) of A211000.

Original entry on oeis.org

3, 15, 31, 35, 43, 47, 51, 151, 155, 175, 179, 243, 247, 251, 255, 339, 343, 347, 355, 359, 363, 367, 371, 391, 395, 403, 407, 6571, 6575, 6583, 6587, 6591, 6595, 6599, 6603, 6607, 6611, 6615, 6619, 6623, 6627, 6631, 6635, 6655, 6659, 6675, 6679, 6683, 6687

Offset: 1

Omar E. Pol, Mar 30 2012

nonn

The behavior seems to be as modular arithmetic but in a growing structure. For n >= 13 the structure is essentially a column of tangent circles of radius 1.

Cf. A211000, A211001, A211003, A211010, A211011, A211020-211023.

A211022 Value on the axis "y" of the center of the n-th circle formed in the structure of A211000.

Original entry on oeis.org

-6, -4, -2, 0, 2, 6, 8, 4, -10, -12, -8, 10, 12, 14, 16, 20, 22, 24, 26, 30, 32, 34, 36, 28, 38, 40, 42, 46, 48, 52, 56, 58, 60, 62, 54, 50, 44, 64, 66, 68, 18, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 106, 104, 100

Offset: 1

Omar E. Pol, Mar 31 2012

sign

The structure of A211000 looks like essentially a column of tangent circles of radius 1. The structure arises from the prime numbers A000040. Values on the axis "x" are equal to 3.

Cf. A187210, A211000, A211001-A211003, A211010, A211011, A211020, A211021, A211023, A211024.

cp's OEIS Frontend

A211000 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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A211020 Number of circles in the structure of A211000 after n-th stage.

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A211011 Value on the axis "y" of the endpoint of the structure (or curve) of A211000 at n-th stage.

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A211010 Value on the axis "x" of the endpoint of the structure of A211000 at n-th stage.

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A211003 Primes congruent to 3 in the structure (or curve) of A211000.

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A211023 Value on the axis "y" of the endpoint of the structure of A211000 if the index is prime.

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A211002 Numbers congruent to 3 in the structure (or curve) of A211000.

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A211022 Value on the axis "y" of the center of the n-th circle formed in the structure of A211000.

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