A070201 -id:A070201 - cp's OEIS frontend

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

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

Clark Kimberling

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Right Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A005044, A070093, A070101.

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

A070204 Number of isosceles 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, 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, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070201(n) - A070203(n).

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Isosceles Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070139, A059169.

A070209 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer inradius.

Original entry on oeis.org

17, 116, 212, 269, 368, 370, 493, 561, 587, 659, 850, 1204, 1297, 1582, 1599, 1629, 1920, 1988, 2115, 2352, 2555, 2574, 2774, 2778, 3251, 3473, 3746, 3751, 4286, 4298, 4307, 4313, 4319, 4330, 4370, 4406, 5008, 5251

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(3)=212: [A070080(212), A070081(212), A070082(212)] = [5,12,13], for s = A070083(212)/2 = (5+12+13)/2 = 15: inradius = sqrt((s-5)*(s-12)*(s-13)/s) = sqrt(10*3*2/15) = sqrt(4) = 2; therefore A070200(212)=2. [Corrected by _Rick L. Shepherd_, May 15 2008]

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.

Eric Weisstein's World of Mathematics, Incircle.
R. Zumkeller, Integer-sided triangles

Cf. A070142, A070201, A051516.

A336757 Number of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression with a perimeter = n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 5, 0, 0, 2, 0, 0, 6, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 8, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 6, 0, 0, 5, 0, 0, 11, 0, 0, 4

Offset: 1

Bernard Schott, Sep 20 2020

nonn

Equivalently: number of primitive integer-sided triangles such that b = (a+c)/2 with a < c and perimeter = n.
As the perimeter of these triangles = 3*b where b is the middle side, a(n) >= 1 iff n = 3*b, with b >= 3.
When b is prime, all the triangles of perimeter n = 3*b are primitive, hence in this case: a(n) = A024164(n).
For the corresponding triples (primitive or not), miscellaneous properties and references, see A336750.

			a(9) = 1 for the smallest such triangle (2, 3, 4).
a(12) = 1 for the Pythagorean triple (3, 4, 5).
a(15) = 2 for the two triples (3, 5, 7) and (4, 5, 6).
a(18) = 1 for the triple (5, 6, 7); the other triple (4, 6, 8) corresponding to a perimeter = 18 is not a primitive triple.

Cf. A336750 (triples, primitive or not), A336755 (primitive triples), A336756 (perimeters of primitive triangles).
Cf. A024164 (number of such triangles, primitive or not).
Similar sequences: A005044 (integer-sided triangles), A024155 (right triangles), A070201 (with integral inradius).
Cf. A000010, A023022.

For n = 3*b, b >= 3, a(n) = A023022(b) = A000010(b)/2, otherwise a(n) = 0.

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

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070201(n) - A070204(n).

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Scalene Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A024153, A005044.

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

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

Reinhard Zumkeller, May 05 2002

nonn

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.

Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Acute Triangle.
R. Zumkeller, Integer-sided triangles

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

a(n) = A070201(n) - A024155(n) - A070206(n).

A070206 Number of obtuse 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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070201(n) - A024155(n) - A070205(n).

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.

Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Obtuse Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070141, A070101.

A070210 Inradii of integer triangles [A070080(A070209(n)), A070081(A070209(n)), A070082(A070209(n))].

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 2, 4, 3, 4, 4, 3, 2, 4, 5, 3, 6, 4, 6, 6, 6, 4, 6, 3, 4, 3, 6, 4, 5, 4, 3, 6, 5, 7, 8, 6, 4, 6, 8, 7, 8, 9, 3, 9, 5, 6, 9, 8, 10, 6, 6, 6, 9, 8, 4, 8, 9, 7, 10, 6, 10, 12, 6, 12, 12, 5, 3, 7, 8, 10, 4, 9, 10, 11, 6, 12, 3, 6, 9, 12, 12, 7, 8

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070200(A070209(n)).

			A070209(3)=212: [A070080(212), A070081(212), A070082(212)] = [5,12,13], let s = A070083(212)/2 = (5+12+13)/2 = 15 then inradius = sqrt((s-5)*(s-5)*(s-6)/s) = sqrt(10*3*2/15) = sqrt(4) = 2; a(3) = A070200(212) = 2.

Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.
Eric Weisstein's World of Mathematics, Incircle.
R. Zumkeller, Integer-sided triangles

Cf. A070149, A070201.

A070208 Number of integer triangles with perimeter n having integral area but not integral inradius.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051516(n) - A070201(n).

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.

Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Heron's Formula.
R. Zumkeller, Integer-sided triangles

A338393 Smallest perimeter of integer-sided triangles for which there exist exactly n triangles that have an integer inradius.

Original entry on oeis.org

12, 36, 60, 162, 108, 180, 228, 84, 132, 168, 210, 640, 252, 448, 504, 612, 462, 480, 396, 1050, 1008, 630, 672, 1632, 756, 792, 1380, 420, 1740, 1232, 1584, 1560, 1188, 1540, 2052, 1428, 1820, 840, 1620, 1320, 1890, 3612, 2912, 2280, 1092, 924, 2340, 2730, 3220

Offset: 1

Bernard Schott, Oct 28 2020

nonn

			a(1) = 12 because (3,4,5) is the smallest integer-sided triangle with an integer inradius and this integer radius = 1.
a(2) = 36 and the 2 corresponding triangles are (9,10,17) with r=2 and (9,12,15) with r=3.
a(3) = 60 and the 3 corresponding triangles are (6,25,29) with r=2, (10,24,26) with r=4 and (15,20,25) with r=5.

Eric Weisstein's World of Mathematics, Incircle.

Cf. A005044, A070201, A120062.

More terms from Amiram Eldar, Oct 28 2020

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A070204 Number of isosceles integer triangles with perimeter n having integral inradius.

Views

Author

Keywords

Comments

Links

Crossrefs

A070209 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer inradius.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A336757 Number of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression with a perimeter = n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A070203 Number of scalene integer triangles with perimeter n having integral inradius.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Ruby

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

Views

Author

Keywords

References

Links

Crossrefs

Formula

A070206 Number of obtuse integer triangles with perimeter n having integral inradius.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A070210 Inradii of integer triangles [A070080(A070209(n)), A070081(A070209(n)), A070082(A070209(n))].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A070208 Number of integer triangles with perimeter n having integral area but not integral inradius.

Views

Author

Keywords

Comments

References

Links

A338393 Smallest perimeter of integer-sided triangles for which there exist exactly n triangles that have an integer inradius.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions