cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Omar E. Pol, Mar 30 2012

Keywords

Comments

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.

Examples

			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)

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

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

Views

Author

Omar E. Pol, Mar 30 2012

Keywords

Comments

For the values on the axis "y" see A211011.
Also [0, 1] together with the absolute values of A131731.

Links

Crossrefs

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

Programs

Formula

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

Views

Author

Omar E. Pol, Mar 30 2012

Keywords

Comments

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.

Crossrefs

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

Views

Author

Omar E. Pol, Mar 30 2012

Keywords

Comments

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.

Crossrefs

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

Views

Author

Omar E. Pol, Mar 31 2012

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

a(n) = A211011(A000040(n)).
Showing 1-5 of 5 results.

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