A211000 -id:A211000 - cp's OEIS frontend

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

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.

A211001 Numbers congruent to 2 in the structure of A211000.

Original entry on oeis.org

2, 34, 42, 50, 150, 242, 246, 250, 354, 358, 370, 390, 394, 402, 406, 6570, 6602, 6606, 6622, 6626, 6630, 6634, 6654, 6658, 6682, 6686

Offset: 1

Omar E. Pol, Mar 30 2012

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 arises from the prime numbers A000040. For the number of circles at n-th stage see A211020.

Cf. A187210, A210838, A210841, A211000, A211002, A211003, A211010, A211011, A211020-A211023.

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)).

A211021 Numbers n such that a new circle appears in the structure of A211000.

Original entry on oeis.org

11, 13, 15, 34, 41, 65, 71, 75, 275, 281, 285, 437, 443, 561, 567, 575, 581, 591, 597, 605, 611, 617, 647, 663, 957, 971, 1025, 1037, 1043, 1055, 1067, 1073, 1091, 1113, 1153, 1165, 1711, 2243, 3377, 3467, 5809, 7937, 7955, 8021, 8043, 8057, 8063

Offset: 1

Omar E. Pol, Mar 31 2012

nonn

Also where the positive records occur in A211020.

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 arises from the prime numbers A000040.

Paolo Xausa, Table of n, a(n) for n = 1..250
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A187210, A211000-A211003, A211010, A211011, A211020, A211022-A211024.

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}]]];
Flatten[Position[Differences[A211020[1000]],1]] (* Paolo Xausa, Jan 16 2023 *)

A211024 Sum of all visible nodes in the structure of A211000 at n-th stage.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 59, 71, 79, 93, 105, 117, 121, 133, 141, 153, 165, 177, 181, 193, 201, 209, 213, 217, 221, 237, 253, 285, 318, 350, 354, 358, 362, 400, 439, 479, 483, 491, 499, 527, 543, 559, 563, 575, 583, 591, 595, 599, 603

Offset: 0

Omar E. Pol, Apr 14 2012

nonn

First differs from A000217 at a(11). For n >= 13 the Q-toothpick 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.

			Consider the illustration of the nodes in structure of A211000:
-----------------------------------------------------
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
.
----------------------------------------------------
We can see that:
a(9)  = 0+1+2+3+4+5+6+7+8+9 = a(8)+9 = 45
a(10) = 0+1+2+3+4+5+6+7+8+9+10 = a(9)+10 = 55
a(11) = 0+1+2+3+4+5+6+8+9+10+11 = a(10)-7+11 = 59

Paolo Xausa, Table of n, a(n) for n = 0..9999 (terms 0..4999 from Carole Dubois)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000217, A187210, A211000-A211003, A211010, A211011, A211020-A211023.

A211024[nmax_]:=Module[{ep={0,0},node=Association[],angle=3/4Pi,turn=Pi/2},Join[{0},Table[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];ep=AngleVector[ep,{Sqrt[2],angle}];node[ep]=n+1;Total[node],{n,0,nmax-1}]]];
A211024[100] (* Paolo Xausa, Jan 16 2023 *)

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

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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A211001 Numbers congruent to 2 in the structure 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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A211021 Numbers n such that a new circle appears in the structure of A211000.

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A211024 Sum of all visible nodes in the structure of A211000 at n-th stage.

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

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