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.

A024155 Number of integer-sided triangles with sides a,b,c, a

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Also number of right integer triangles with perimeter n having integral inradius. - Reinhard Zumkeller, May 05 2002
Every integer-sided right triangle has integer inradius. If the triple is [p^2-q^2,2pq,p^2+q^2] then inradius = pq-q^2. - Michael Somos, Sep 13 2005

Links

Crossrefs

Cf. A005044, A070093, A070101.

Formula

a(n) = A070201(n) - A070205(n) - A070206(n).

A070201 Number of integer triangles with perimeter n having integral inradius.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 8, 0, 0, 0, 1, 0, 3
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

a(n) = #{k | A070083(k) = n and A070200(k) = exact inradius};
a(n) = A070203(n) + A070204(n);
a(n) = A070205(n) + A070206(n) + A024155(n);
a(odd) = 0.

Examples

			a(36)=2, as there are two integer triangles with integer inradius having perimeter=32:
First: [A070080(368), A070081(368), A070082(368)] = [9,10,17], for s = A070083(368)/2 = (9+10+17)/2 = 18: inradius = sqrt((s-9)*(s-10)*(s-17)/s) = sqrt(9*8*1/18) = sqrt(4) = 2; therefore A070200(368) = 2.
2nd: [A070080(370), A070081(370), A070082(370)] = [9,12,15], for s = A070083(370)/2 = (9+12+15)/2 = 18: inradius = sqrt((s-9)*(s-12)*(s-15)/s) = sqrt(9*6*3/18) = sqrt(9) = 3; therefore A070200(370) = 3.

Links

Crossrefs

Cf. A070209, A070202, A070208, A005044, A070140.
Cf. A120062, A120572, A331040.

Programs

  • Ruby
    def A(n)
  cnt = 0
  (1..n / 3).each{|a|
    (a..(n - a) / 2).each{|b|
      c = n - a - b
      if a + b > c
        s = n / 2r
        t = (s - a) * (s - b) * (s - c) / s
        if t.denominator == 1
          t = t.to_i
          cnt += 1 if Math.sqrt(t).to_i ** 2 == t
        end
      end
    }
  }
  cnt
end
def A070201(n)
  (1..n).map{|i| A(i)}
end
p A070201(100) # Seiichi Manyama, Oct 06 2017

A070205 Number of acute integer triangles with perimeter n having integral inradius.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

References

  • Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Links

Crossrefs

Cf. A024155, A070093, A070140, A070201, A070206.

Formula

a(n) = A070201(n) - A024155(n) - A070206(n).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.