cp's OEIS Frontend

A156689 Inradii of primitive Pythagorean triples a^2+b^2=c^2, 0 with gcd(a,b)=1, and sorted to correspond to increasing a (given in A020884).

%I A156689 #13 Jul 07 2016 23:48:48
%S A156689 1,2,3,3,4,5,5,6,7,7,8,9,6,9,10,11,11,12,13,10,13,14,15,15,12,16,17,
%T A156689 14,17,18,15,19,19,20,21,18,21,22,23,15,23,24,21,25,22,25,26,27,27,24,
%U A156689 28,29,21,26,29,30
%N A156689 Inradii of primitive Pythagorean triples a^2+b^2=c^2, 0<a<b<c with gcd(a,b)=1, and sorted to correspond to increasing a (given in A020884).
%C A156689 The inradius is given by r=1/2 (a+b-c)=ab/(a+b+c)=area/semiperimeter, and the inradii ordered by increasing r are in A020888.
%D A156689 Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32.  Solution published in Vol. 16, Issue 2, November 2008, p. 32.
%D A156689 D. G. Rogers, Putting Pythagoras in the frame, Mathematics Today, The Institute of Mathematics and its Applications, Vol. 44, No. 3, June 2008, pp. 123-125.
%H A156689 Reinhard Zumkeller, <a href="/A156689/b156689.txt">Table of n, a(n) for n = 1..1000</a>
%H A156689 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Right-angled Triangles and Pythagoras' Theorem</a>
%F A156689 A156689(n)=1/2 (A020884(n)+A156678(n)-A156679(n))
%e A156689 The eighth primitive Pythagorean triple ordered by increasing a is (13,84,85). As this has inradius 1/2 (13+84-85)=6, we have a(8)=6.
%t A156689 PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i<n,If[GCD[i,j]==1,h=Sqrt[i^2+j^2];If[IntegerQ[h] && j<n,AppendTo[t,{i,j,h}]];];If[j<n,j+=2,i++;j=i+1]];t];k=30;data1=PrimitivePythagoreanTriplets[2k^2+2k+1];data2=Select[data1,#[[1]]<=2k+1 &];1/2(#[[1]]+#[[2]]-#[[3]]) &/@data2
%o A156689 (Haskell)
%o A156689 a156689 n = a156689_list !! (n-1)
%o A156689 a156689_list = f 1 1 where
%o A156689    f u v | v > uu `div` 2        = f (u + 1) (u + 2)
%o A156689          | gcd u v > 1 || w == 0 = f u (v + 2)
%o A156689          | otherwise             = (u + v - w) `div` 2 : f u (v + 2)
%o A156689          where uu = u ^ 2; w = a037213 (uu + v ^ 2)
%o A156689 -- _Reinhard Zumkeller_, Nov 09 2012
%Y A156689 Cf. A020884, A020888, A156678, A156679.
%Y A156689 Cf. A037213.
%K A156689 easy,nice,nonn
%O A156689 1,2
%A A156689 _Ant King_, Feb 18 2009