A070201 - OEIS

%I #28 Oct 29 2020 03:27:34

%S 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 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 0,2,0,2,0,0,0,1,0,1,0,2,0,0,0,8,0,0,0,1,0,3

%N Number of integer triangles with perimeter n having integral inradius.

%C a(n) = #{k | A070083(k) = n and A070200(k) = exact inradius};

%C a(n) = A070203(n) + A070204(n);

%C a(n) = A070205(n) + A070206(n) + A024155(n);

%C a(odd) = 0.

%H Seiichi Manyama, <a href="/A070201/b070201.txt">Table of n, a(n) for n = 1..5000</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Incircle.html">Incircle</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronsFormula.html">Heron's Formula</a>.

%H Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>

%e a(36)=2, as there are two integer triangles with integer inradius having perimeter=32:

%e 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 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 (Ruby)

%o def A(n)

%o   cnt = 0

%o   (1..n / 3).each{|a|

%o     (a..(n - a) / 2).each{|b|

%o       c = n - a - b

%o       if a + b > c

%o         s = n / 2r

%o         t = (s - a) * (s - b) * (s - c) / s

%o         if t.denominator == 1

%o           t = t.to_i

%o           cnt += 1 if Math.sqrt(t).to_i ** 2 == t

%o         end

%o       end

%o     }

%o   }

%o   cnt

%o end

%o def A070201(n)

%o   (1..n).map{|i| A(i)}

%o end

%o p A070201(100) # _Seiichi Manyama_, Oct 06 2017

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

%Y Cf. A120062, A120572, A331040.

%K nonn

%O 1,36

%A _Reinhard Zumkeller_, May 05 2002