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 1-4 of 4 results.

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))

A246919 The length of the longest 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, 19, 14, 37, 13, 61, 38, 91, 28, 127, 74, 169, 49, 217, 122, 271, 76, 331, 182, 397, 109, 469, 254, 547, 148, 631, 338, 721, 193, 817, 434, 919, 244, 1027, 542, 1141, 301, 1261, 662, 1387, 364, 1519, 794, 1657, 433, 1801, 938, 1951, 508, 2107
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, A246918, A246920.

Programs

  • Mathematica
    Rest@ CoefficientList[Series[x^3 (7 + 19 x^2 + 14 x^3 + 16 x^4 + 13 x^5 + 4 x^6 - 4 x^7 + x^8 - 11 x^9 + x^10 + 2 x^11 + 4 x^13)/((1 - x)^3 (1 + x)^3 (1 + x^2)^3), {x, 0, 53}], x] (* Michael De Vlieger, Jun 06 2016 *)
  • PARI
    \\ Returns the length of the longest integral cevian of an equilateral triangle of side n.
longest(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)
      )
    )
  );
  if(#s==1, return(0));
  for(i=1, #s, if(s[i]!=n, return(s[i])))
}
vector(100, n, longest(n))

Formula

Conjectures from Colin Barker, Jun 06 2016: (Start)
a(n) = 3*a(n-4)-3*a(n-8)+a(n-12) for n>14.
G.f.: x^3*(7 +19*x^2 +14*x^3 +16*x^4 +13*x^5 +4*x^6 -4*x^7 +x^8 -11*x^9 +x^10 +2*x^11 +4*x^13) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3).
(End)

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)

A229838 Consider all primitive 60-degree triangles with sides A < B < C. The sequence gives the values of A.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 104, 105
Offset: 1

Views

Author

Colin Barker, Oct 01 2013

Keywords

Comments

A primitive triangle is one for which the sides have no common factor.
A004611 gives the values of B, and A089025 gives the values of C.

Examples

			7 appears in the sequence because there exists a primitive 60-degree triangle with sides 7, 37 and 40.

Links

Crossrefs

Cf. A004611, A089025, A229839.

Programs

  • PARI
    \\ Gives terms not exceeding amax
\\ e.g. pt60a(25) gives [3,5,7,8,9,11,13,15,16,17,19,21,23,24,25]
pt60a(amax) = {
  s=[];
  for(m=1, amax\2,
    for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        if(2*m*n+n*n<=amax, s=concat(s, 2*m*n+n*n));
        if(m*m-n*n<=amax, s=concat(s, m*m-n*n))
      )
    )
  );
  vecsort(s,,8)
}

Formula

Empirical g.f.: -x*(x^5-x^4-x^3-2*x^2-2*x-3) / ((x-1)^2*(x^4+x^3+x^2+x+1)).
Showing 1-4 of 4 results.

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