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