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

%I A070201 #29 Feb 16 2025 08:32:46
%S A070201 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,
%T A070201 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,
%U A070201 0,2,0,2,0,0,0,1,0,1,0,2,0,0,0,8,0,0,0,1,0,3
%N A070201 Number of integer triangles with perimeter n having integral inradius.
%C A070201 a(n) = #{k | A070083(k) = n and A070200(k) = exact inradius};
%C A070201 a(n) = A070203(n) + A070204(n);
%C A070201 a(n) = A070205(n) + A070206(n) + A024155(n);
%C A070201 a(odd) = 0.
%H A070201 Seiichi Manyama, <a href="/A070201/b070201.txt">Table of n, a(n) for n = 1..5000</a>
%H A070201 Mohammad K. Azarian, <a href="http://www.jstor.org/stable/25678790">Solution to Problem S125: Circumradius and Inradius</a>, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
%H A070201 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Incircle.html">Incircle</a>.
%H A070201 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronsFormula.html">Heron's Formula</a>.
%H A070201 Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%e A070201 a(36)=2, as there are two integer triangles with integer inradius having perimeter=32:
%e A070201 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.
%e A070201 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.
%o A070201 (Ruby)
%o A070201 def A(n)
%o A070201   cnt = 0
%o A070201   (1..n / 3).each{|a|
%o A070201     (a..(n - a) / 2).each{|b|
%o A070201       c = n - a - b
%o A070201       if a + b > c
%o A070201         s = n / 2r
%o A070201         t = (s - a) * (s - b) * (s - c) / s
%o A070201         if t.denominator == 1
%o A070201           t = t.to_i
%o A070201           cnt += 1 if Math.sqrt(t).to_i ** 2 == t
%o A070201         end
%o A070201       end
%o A070201     }
%o A070201   }
%o A070201   cnt
%o A070201 end
%o A070201 def A070201(n)
%o A070201   (1..n).map{|i| A(i)}
%o A070201 end
%o A070201 p A070201(100) # _Seiichi Manyama_, Oct 06 2017
%Y A070201 Cf. A070209, A070202, A070208, A005044, A070140.
%Y A070201 Cf. A120062, A120572, A331040.
%K A070201 nonn
%O A070201 1,36
%A A070201 _Reinhard Zumkeller_, May 05 2002