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 1-3 of 3 results.

A229839 Consider all 60-degree triangles with sides A < B < C. The sequence gives the values of C.

Original entry on oeis.org

8, 15, 16, 21, 24, 30, 32, 35, 40, 42, 45, 48, 55, 56, 60, 63, 64, 65, 70, 72, 75, 77, 80, 84, 88, 90, 91, 96, 99, 104, 105, 110, 112, 117, 119, 120, 126, 128, 130, 133, 135, 136, 140, 143, 144, 147, 150, 152, 153, 154, 160, 165, 168, 171, 175, 176, 180, 182
Offset: 1

Views

Author

Colin Barker, Oct 01 2013

Keywords

Comments

A009005 gives the values of A, and A050931 gives the values of B.
The side n of an equilateral triangle for which a nontrivial integral cevian of length less than n exists, which divides an edge into two integral parts. - Colin Barker, Sep 09 2014

Examples

			16 appears in the sequence because there exists a 60-degree triangle with sides 6, 14 and 16.

Links

Crossrefs

Cf. A009005, A050931, A229838.
Cf. A246918, A246919, A246920.

Programs

  • Mathematica
    list={};cmax=182;
Do[If[IntegerQ[Sqrt[e^2-e t+t^2]],AppendTo[list,e]],{e,2,cmax},{t,1,e-1}]
list//DeleteDuplicates (* Herbert Kociemba, Apr 25 2021 *)
  • PARI
    \\ Gives values of C not exceeding cmax.
\\ e.g. t60c(60) gives [8, 15, 16, 21, 24, 30, 32, 35, 40, 42, 45, 48, 55, 56, 60]
t60c(cmax) = {
  v=pt60c(cmax);
  s=[];
  for(i=1, #v,
    for(m=1, cmax\v[i],
      if(v[i]*m<=cmax, s=concat(s, v[i]*m))
    )
  );
  vecsort(s,,8)
}
\\ Gives values of C not exceeding cmax in primitive triangles.
\\ e.g. pt60c(115) gives [8, 15, 21, 35, 40, 48, 55, 65, 77, 80, 91, 96, 99, 112]
pt60c(cmax) = {
  s=[];
  for(m=1, ceil(sqrt(cmax+1)),
   for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        if(2*m*n+m*m<=cmax, s=concat(s, 2*m*n+m*m))
      )
    )
  );
  vecsort(s,,8)
}

A246918 The length of the shortest nontrivial integral cevian of an equilateral triangle of side n that divides an edge into two integral parts, or 0 if no such cevian exists.

Original entry on oeis.org

0, 0, 7, 0, 7, 14, 13, 7, 21, 14, 31, 28, 43, 26, 13, 14, 73, 42, 91, 28, 19, 62, 133, 21, 35, 86, 63, 52, 211, 26, 241, 28, 37, 146, 31, 84, 343, 182, 49, 35, 421, 38, 463, 124, 39, 266, 553, 42, 91, 70, 79, 172, 703, 126, 49, 49, 97, 422, 871, 52, 931, 482
Offset: 1

Views

Author

Colin Barker, Sep 07 2014

Keywords

Comments

A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension).
A nontrivial cevian is one that does not coincide with a side of the triangle.
For an equilateral triangle of side n, the lengths of its cevians are the values of y in the solutions to x^2-y^2-n*x+n^2=0.

Links

Crossrefs

Cf. A229839, A246919, A246920.

Programs

  • PARI
    \\ Returns the length of the shortest integral cevian of an equilateral triangle of side n.
shortest(n) = {
  s=[];
  m=12*n^2;
  fordiv(m, f,
    g=m\f;
    if(f<=g && (f+g)%2==0,
      x=(f+g)\2;
      if(x%4==0,
        s=concat(s, x\4)
      )
    )
  );
  s=Colrev(s)~;
  if(#s==1, return(0));
  for(i=1, #s, if(s[i]!=n, return(s[i])))
}
vector(100, n, shortest(n))

A246920 The number of distinct lengths of nontrivial integral cevians of an equilateral triangle of side n that divide an edge into two integral parts.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 5, 4, 2, 2, 2, 2, 5, 2, 2, 5, 4, 2, 3, 2, 2, 5, 2, 6, 5, 2, 8, 2, 2, 2, 5, 8, 2, 5, 2, 2, 8, 2, 2, 9, 4, 4, 5, 2, 2, 3, 8, 8, 5, 2, 2, 5, 2, 2, 8, 8, 8, 5, 2, 2, 5, 8, 2, 8, 2, 2, 9, 2, 8, 5, 2, 14, 4, 2, 2, 5, 8, 2
Offset: 1

Views

Author

Colin Barker, Sep 07 2014

Keywords

Comments

A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension).
A nontrivial cevian is one that does not coincide with a side of the triangle.
For an equilateral triangle of side n, the lengths of its cevians are the values of y in the solutions to x^2-y^2-n*x+n^2=0.

Examples

			a(15) = 5 because cevians of an equilateral triangle of side 15 have length 13, 21, 35, 57 or 169.

Links

Crossrefs

Cf. A229839, A246918, A246919.

Programs

  • PARI
    \\ Returns the number of cevians of an equilateral triangle of side n.
count(n) = {
  s=[];
  n=12*n^2;
  fordiv(n, f,
    g=n\f;
    if(f<=g && (f+g)%2==0,
      x=(f+g)\2;
      if(x%4==0,
        s=concat(s, x\4)
      )
    )
  );
  Colrev(s)~
}
vector(100, n, #count(n)-1)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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