A331040 -id:A331040 - cp's OEIS frontend

A331041 Denominator of squared radius of inscribed circles of triangles with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Numerators are A331040.

12, 52, 4, 4, 1, 4, 1, 4, 4, 4, 4, 4, 4, 1, 4, 4, 1, 4, 4, 4, 4, 1, 1, 4, 1

Offset: 1

Hugo Pfoertner, Jan 11 2020

See A331040. Conjecturally, all terms a(n) are either 1 or 4 for n >= 3.

			See A331040.

Cf. A331040, A331042, A331043.

A331043 a(n) is the number of triangles with integer sides i <= j <= k with squared radius of incircle b(n) = A331040(n)/A331041(n). Records of numbers of distinct triangles such that all smaller radii produce fewer triangles sharing the same radius of incircle than the current radius b(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 13, 14, 20, 24, 42, 45, 50, 68, 72, 84, 88, 101, 120, 149, 175, 181, 189, 206, 243, 289

Offset: 1

Hugo Pfoertner, Jan 11 2020

See A331040 for more information and examples.

Cf. A120062, A331040, A331041, A331042.

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

Reinhard Zumkeller, May 05 2002

nonn

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.

			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.

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, Heron's Formula.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070209, A070202, A070208, A005044, A070140.

Cf. A120062, A120572, A331040.

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

A331042 a(n) = 4 * squared radius of inscribed circles of triangles with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. If this radius is not a multiple of (1/4), a(n) = 0.

Original entry on oeis.org

0, 0, 3, 7, 12, 15, 32, 35, 55, 63, 95, 119, 135, 224, 231, 255, 320, 351, 455, 495, 855, 864, 896, 1071, 1440

Offset: 1

Hugo Pfoertner, Jan 11 2020

It is conjectured that all radii of incircles leading to records with the exception of the first two terms are multiples of 1/4, thus a(n) > 0 for all n > 2.

See A331040 for more information and examples.

Cf. A120062, A331012.

Cf. A331040, A331041, A331043 (records of numbers of triangles).

If A331041(n) equals 1 or 4, a(n) = 4 * A331040(n)/A331041(n), 0 otherwise.

A331224 Numerator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Denominators are A331225.

Original entry on oeis.org

1, 64, 49, 1024, 2025, 4096, 25600, 2401, 7744, 148225, 8281, 2073600, 123904, 774400, 3705625

Offset: 1

Hugo Pfoertner, Jan 14 2020

			Correspondence of the first terms b(n) = a(n)/A331225(n) with triangles (i, j, k):
b(1) = 1/3: (1,1,1), start with 1 = A331226(1) triangle.
b(2) = 64/15: (2,3,4), (2,4,4) is the first occurrence of 2 = A331226(2) triangles with identical R.
b(3) = 49/3: (3,5,7), (3,7,8), (5,7,8), (7,7,7) is the first occurrence of more triangles with identical R than the previous record 2, new record is 4 = A331226(3).
b(4) = 1024/15: (5,8,12), (5,14,16), (8,8,14), (8,12,16), (8,16,16), (12,14,16) is the first occurrence of more triangles with identical R than the previous record 4, new record is 6 = A331226(4).

Cf. A331040, A331041, A331225, A331226, A331227, A331228.

Squared radius of circumcircle of triangle with sides a, b, c:

R^2 = (a*b*c)^2 / (16*s*(s - a)*(s - b)*(s - c)) with s = (a + b + c)/2.

A331225 Denominator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Numerators are A331224.

Original entry on oeis.org

3, 15, 3, 15, 11, 15, 39, 3, 7, 96, 3, 119, 7, 39, 96

Offset: 1

Hugo Pfoertner, Jan 14 2020

			See A331224.

Cf. A331040, A331041, A331224, A331226, A331227, A331228.

A331226 a(n) is the number of triangles with integer sides i <= j <= k with squared radius of circumscribed circle b(n) = A331224(n)/A331225(n). Records of numbers of distinct triangles such that all smaller radii produce fewer triangles sharing the same radius of circumcircle than the current radius b(n).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 13, 17, 22, 31, 33, 46, 53, 67

Offset: 1

Hugo Pfoertner, Jan 14 2020

			See A331224.

Cf. A331040, A331041, A331043, A331224, A331225, A331227, A331228.

cp's OEIS Frontend

A331041 Denominator of squared radius of inscribed circles of triangles with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Numerators are A331040.

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A331043 a(n) is the number of triangles with integer sides i <= j <= k with squared radius of incircle b(n) = A331040(n)/A331041(n). Records of numbers of distinct triangles such that all smaller radii produce fewer triangles sharing the same radius of incircle than the current radius b(n).

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

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A331042 a(n) = 4 * squared radius of inscribed circles of triangles with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. If this radius is not a multiple of (1/4), a(n) = 0.

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A331224 Numerator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Denominators are A331225.

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A331225 Denominator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Numerators are A331224.

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