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A070201 Number of integer triangles with perimeter n having integral inradius. 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,36

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.

LINKS

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

EXAMPLE

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.

PROG

(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

CROSSREFS

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

Cf. A120062, A120572, A331040.

Sequence in context: A135767 A208575 A070203 * A070138 A024153 A341523

Adjacent sequences:  A070198 A070199 A070200 * A070202 A070203 A070204

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 05 2002

STATUS

approved

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Last modified March 6 04:42 EST 2021. Contains 341842 sequences. (Running on oeis4.)