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.

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

A070203 Number of scalene 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, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 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) = A070201(n) - A070204(n).

Links

Crossrefs

Cf. A024153, A005044.

Programs

  • Ruby
    def A(n)
  cnt = 0
  (1..n / 3).each{|a|
    (a + 1..(n - a) / 2).each{|b|
      c = n - a - b
      if a + b > c && 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 A070203(n)
  (1..n).map{|i| A(i)}
end
p A070203(100) # Seiichi Manyama, Oct 07 2017

A362670 Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 87, 88, 90, 92, 93, 96, 99, 100, 102, 104, 105, 108, 111, 112, 114, 116, 117, 120, 123, 124, 126, 128, 129, 132, 135
Offset: 1

Views

Author

Bernard Schott, May 05 2023

Keywords

Comments

The inradius for isosceles triangle (a, a, c) is r = (c/2)*sqrt((2*a-c)/(2*a+c)).
If m is a term, so is k*m with k > 1.
As r = 3 and r = 4 are terms, A008585 and A008586 are respective subsequences; the only terms < 100 that are not multiples of 3 or 4 are 35 and 70, the next one is r = 154 = 2*7*11 for triple (765, 765, 1386).
By the triangle inequality, a+1 <= c <= 2*a-1.
Differs from A059267. Examples: 154 is not in A059267 but in this sequence at radius r=154 with side lengths c=1386 and a=765. 442 is not in A059267 but in this sequences with r=442, c=6630, a=3435. - R. J. Mathar, Jun 26 2023

Examples

			The smallest inradius r = 3 corresponds to isosceles triangle (10, 10, 12).
The second inradius r = 4 corresponds to isosceles triangle (15, 15, 24).
r = 15 is the first inradius for which there exist two such isosceles triangles: (50, 50, 60) and (68, 68, 120).
r = 35 is the smallest inradius that is not multiple of 3 or of 4, this inradius corresponds to isosceles triangle (222, 222, 420).

Links

Crossrefs

Cf. A008585, A008586, A070204, A120062, A120570.
Cf. A362669 (similar but with (a,b,b)).

A362669 Integer inradii for which there exists an isosceles triangle with integer sides (a, b, b) where a < b.

Original entry on oeis.org

10, 20, 21, 24, 30, 36, 40, 42, 48, 50, 55, 60, 63, 70, 72, 78, 80, 84, 90, 96, 100, 105, 108, 110, 112, 120, 126, 130, 136, 140, 144, 147, 150, 156, 160, 165, 168, 170, 171, 180, 189, 190, 192, 195, 200, 210, 216, 220, 224, 230, 231, 234, 240, 250, 252, 253, 260, 264, 270, 272, 273, 275
Offset: 1

Views

Author

Bernard Schott, Apr 29 2023

Keywords

Comments

The inradius for isosceles triangle (a, b, b) is r = (a/2)*sqrt((2*b-a)/(2*b+a)).
If m is a term, so is k*m with k > 1; hence, A008592 \ {0} is a subsequence.

Examples

			The smallest inradius, r = 10, corresponds to isosceles triangle (30, 39, 39).
The third inradius, r = 21, corresponds to isosceles triangle (56, 100, 100).
r = 60 is the first inradius for which there exist two such isosceles triangles: (168, 259, 259) and (180, 234, 234).

Links

Crossrefs

Cf. A008592, A070204, A120062, A120570, A362670 (similar but with (a,a,c)).

Programs

  • Mathematica
    Select[Range[300], Length @ Reduce[#^2 == a^2*(2*b - a)/(4*(2*b + a)) && 0 < a < b, {a, b}, Integers] > 0 &] (* Amiram Eldar, May 05 2023 *)
Showing 1-4 of 4 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.