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.

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A357434 a(n) is the number of distinct Q-toothpicks after the n-th stage of the structure described in A211000.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 20, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 23, 24, 25, 26, 27, 28, 28
Offset: 0

Views

Author

Paolo Xausa, Sep 28 2022

Keywords

Comments

See A211000 for additional information.
For the definition of Q-toothpicks, see A187210.

Examples

			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).
.
n       0       1      10      15      32      39      60      65
a(n)    0       1      10      15      16      20      23      28
.
                                                                /\
                                                                \/
                                                                 \
                                                         /       /
                                                /       /\      /\
                                                \       \/      \/
              /       /\      /\      /\      /\/\    /\/\    /\/\
                        \       \       \/      \/      \/      \/
                         \      /\      /\      /\      /\      /\
                         /      \/      \/      \/      \/      \/
                        /       /\      /\      /\      /\      /\
                        \       \/      \/      \/      \/      \/
                         \      /\      /\      /\      /\      /\
                        \/      \/      \/      \/      \/      \/
.

Links

Crossrefs

Cf. A187210, A211000, A355479.

Programs

  • Mathematica
    A357434[nmax_]:=Module[{a={0},tp={},ep1={0,0},ep2,angle=0,turn=Pi/2},Do[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];tp=Union[tp,{{ep1,ep2=AngleVector[ep1,angle]}}];ep1=ep2;AppendTo[a,Length[tp]],{n,0,nmax-1}];a];
A357434[100]
  • PARI
    A357434(nmax) = my(a=List([0,1]), newtp=[[0, 0], [1, 1]], tp=Set([newtp]), turn=1, p1, p2); if(nmax==0, return([0]));for(n=1, nmax-1, p1=newtp[1]; p2=newtp[2]; if(isprime(n), newtp=[p2, [2*p2[1]-p1[1], 2*p2[2]-p1[2]]], if(n>5 && isprime(n-1), turn*=-1); newtp=[p2, [p2[1]-turn*(p1[2]-p2[2]), p2[2]+turn*(p1[1]-p2[1])]]); tp=setunion(tp, [newtp]); listput(a,length(tp))); Vec(a);
A357434(100)
  • Python
    from sympy import isprime
def A357434(nmax):
    newtp, a, turn = ((0, 0), (1, 1)), [0, 1], 1
    tp = {newtp}
    for n in range(1, nmax):
        p1, p2 = newtp[0], newtp[1]
        if isprime(n): # Continue straight
            newtp = (p2, (2*p2[0]-p1[0], 2*p2[1]-p1[1]))
        else:          # Turn
            if n>5 and isprime(n-1): turn *= -1
            newtp = (p2, (p2[0]-turn*(p1[1]-p2[1]), p2[1]+turn*(p1[0]-p2[0])))
        tp.add(newtp)
        a.append(len(tp))
    return a[:nmax+1]
print(A357434(100))

A210838 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks of size = 1..n. The inflection points are the n-th nodes if n is a triangular number A000217.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 0, 6, -4, 10, 1, 15, 7, 9, 14, 2, 22, 10, 13, 19, 3, 9, -8, -2, -20, 10, -7, 23, 7, 9, -8, -6, -24, -22, -7, -39, 11, -21, -8, -2, -28, -22, -7, -43, 15, -65, -8, -88, -32, -64, -7, -39, 19, -65, -8, -92, -36, -64, -65, -35, -95, -65, -64, -96
Offset: 0

Views

Author

Omar E. Pol, Mar 28 2012

Keywords

Comments

It appears there is an infinite family of this type of curves or structures in which the terms of a sequence of positive integers are represented as inflection points and the gaps between them are essentially represented as nodes of spirals. For example: consider a structure formed by Q-toothpicks of size = Axxxxxa connected by their endpoints in which the inflection points are the exposed endpoints at stage Axxxxxb(n), where both Axxxxxa and Axxxxxb are sequences with positive integers. Also instead of Q-toothpicks we can use semicircumferences or also 3/4 of circumferences. For the definition of Q-toothpicks see A187210.
We start at stage 0 with no Q-toothpicks.
At stage 1 we place a Q-toothpick of size 1 centered at (1,0) with its endpoints at (0,0) and (1,1). Since 1 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 2 we place a Q-toothpick of size 2 centered at (1,3) with its endpoints at (1,1) and (3,3).
At stage 3 we place a Q-toothpick of size 3 centered at (0,3) with its endpoints at (3,3) and (0,6). Since 3 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 4 we place a Q-toothpick of size 4 centered at (0,10) with its endpoints at (0,6) and (-4,10).
And so on...

Examples

			-------------------------------------
Stage n also              The end as
the size of     Pair      inflection
Q-toothpick   (x    y)      point
-------------------------------------
.    0         0,   0,        -
.    1         1,   1,       Yes
.    2         3,   3,        -
.    3         0,   6,       Yes
.    4        -4,  10,        -
.    5         1,  15,        -
.    6         7,   9,       Yes
.    7        14,   2,        -
.    8        22,  10,        -
.    9        13,  19,        -
.   10         3,   9,       Yes
.   11        -8,  -2,        -
.   12       -20,  10,        -
.   13        -7,  23,        -
.   14         7,   9,        -
.   15        -8,  -6,       Yes

Links

Crossrefs

Cf. A210841 (the same idea for primes).
Cf. A000217, A005132, A187210, A210606, A211000, A211011.

Programs

  • Mathematica
    A210838[nmax_]:=Module[{ep={0, 0}, angle=3/4Pi, turn=Pi/2, infl=0}, Join[{ep}, Table[If[n>1&&IntegerQ[Sqrt[8(n-1)+1]], infl++, If[Mod[infl, 2]==1, turn*=-1]; angle-=turn; infl=0]; ep=AngleVector[ep, {Sqrt[2]n, angle}], {n, nmax}]]];
A210838[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Jan 12 2023 *)
  • PARI
    A210838(nmax) = my(ep=vector(nmax+1), turn=1, infl=0, ep1, ep2); ep[1]=[0, 0]; if(nmax==0, return(ep)); ep[2]=[1, 1]; for(n=2, nmax, ep1=ep[n-1]; ep2=ep[n]; if(issquare((n-1)<<3+1), infl++; ep[n+1]=[ep2[1]+n*sign(ep2[1]-ep1[1]), ep2[2]+n*sign(ep2[2]-ep1[2])], if(infl%2, turn*=-1); infl=0; ep[n+1]=[ep2[1]-turn*n*sign(ep1[2]-ep2[2]), ep2[2]+turn*n*sign(ep1[1]-ep2[1])])); ep;
A210838(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023
  • Python
    from numpy import sign
from sympy import integer_nthroot
def A210838(nmax):
    ep, turn, infl = [(0, 0), (1, 1)], 1, 0
    for n in range(2, nmax + 1):
        ep1, ep2 = ep[-2], ep[-1]
        if integer_nthroot(((n - 1) << 3) + 1, 2)[1]: # Continue straight
            infl += 1
            dx = n * sign(ep2[0] - ep1[0])
            dy = n * sign(ep2[1] - ep1[1])
        else: # Turn
            if infl % 2: turn *= -1
            infl = 0
            dx = turn * n * sign(ep2[1] - ep1[1])
            dy = turn * n * sign(ep1[0] - ep2[0])
        ep.append((ep2[0] + dx, ep2[1] + dy))
    return ep[:nmax+1]
print(A210838(100)) # Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023

Extensions

a(30)-a(33) corrected and more terms by Paolo Xausa, Jan 12 2023

A210841 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks of size = 1..n. The inflection points are the n-th nodes if n is prime.

Original entry on oeis.org

0, 0, 1, 1, 3, -1, 6, -4, 10, -8, 5, -13, -1, -19, 6, -26, 14, -34, 5, -43, -5, -33, 6, -22, 18, -10, 5, 3, -9, 17, 6, 32, 22, 16, 5, -1, -13, -19, 6, -38, 26, -58, 5, -79, -17, -57, 6, -34, 30, -10, 5, 15, -21, -11, 6, -38, 34, -10, 5, 19, -25, 49, 6, 80, 38, 112
Offset: 0

Views

Author

Omar E. Pol, Mar 29 2012

Keywords

Comments

The same idea as A210838 but here the inflection points are prime numbers.

Examples

			-------------------------------------
Stage n also              The end as
the size of     Pair      inflection
Q-toothpick   (x    y)      point
-------------------------------------
.    0         0,   0,        -
.    1         1,   1,        -
.    2         3,  -1,       Yes
.    3         6,  -4,       Yes
.    4        10,  -8,        -
.    5         5, -13,       Yes
.    6        -1, -19,        -
.    7         6, -26,       Yes

Links

Crossrefs

Cf. A000040, A187210, A210606, A210838, A211000, A211011.

Programs

  • Mathematica
    A210841[nmax_]:=Module[{ep={0,0},angle=3/4Pi,turn=Pi/2},Join[{ep},Table[If[!PrimeQ[n-1],If[n>6&&PrimeQ[n-2],turn*=-1];angle-=turn];ep=AngleVector[ep,{Sqrt[2]n,angle}],{n,nmax}]]];
A210841[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Mar 04 2023 *)
  • PARI
    A210841(nmax) = my(ep=vector(nmax+1), turn=1, ep1, ep2); ep[1]=[0, 0]; if(nmax==0, return(ep)); ep[2]=[1, 1]; for(n=2, nmax, ep1=ep[n-1]; ep2=ep[n]; if(isprime(n-1), ep[n+1]=[ep2[1]+n*sign(ep2[1]-ep1[1]), ep2[2]+n*sign(ep2[2]-ep1[2])], if(n>6 && isprime(n-2), turn*=-1); ep[n+1]=[ep2[1]-turn*n*sign(ep1[2]-ep2[2]), ep2[2]+turn*n*sign(ep1[1]-ep2[1])])); ep;
A210841(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Mar 04 2023
  • Python
    from numpy import sign
from sympy import isprime
def A210841(nmax):
    ep, turn = [(0, 0), (1, 1)], 1
    for n in range(2, nmax + 1):
        ep1, ep2 = ep[-2], ep[-1]
        if isprime(n - 1): # Continue straight
            dx = n * sign(ep2[0] - ep1[0])
            dy = n * sign(ep2[1] - ep1[1])
        else: # Turn
            if n > 6 and isprime(n - 2): turn *= -1
            dx = turn * n * sign(ep2[1] - ep1[1])
            dy = turn * n * sign(ep1[0] - ep2[0])
        ep.append((ep2[0] + dx, ep2[1] + dy))
    return ep[:nmax+1]
print(A210841(100)) # Generates 101 coordinate pairs - Paolo Xausa, Mar 04 2023

Extensions

a(14) corrected by and more terms from Paolo Xausa, Mar 04 2023
Previous Showing 11-13 of 13 results.

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