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%I #73 Oct 21 2019 13:57:43
%S 2,2,3,3,5,6,4,9,10,5,6,8,5,14,15,6,9,11,6,20,21,7,27,28,8,10,13,8,35,
%T 36,9,13,16,9,21,23,9,44,45,10,26,28,10,54,55,11,14,18,11,20,23,11,65,
%U 66,12,17,21,12,24,27
%N Consider triples a<=b<c where (a^2+b^2-c^2)/(c-a-b) =1, ordered by a and then b; sequence gives a, b and c values in that order.
%C The definition can be generalized to define Pythagorean k-triples a<=b<c where (a^2+b^2-c^2)/(c-a-b)=k, or where for some integer k, a(a+k) + b(b+k) = c(c+k).
%C If a, b and c form a Pythagorean k-triple, then na, nb and nc form a Pythagorean nk-triple.
%C A triangle is defined to be a Pythagorean k-triangle if its sides form a Pythagorean k-triple.
%C If a, b and c are the sides of a Pythagorean k-triangle ABC with a<=b<c, then cos(C) = -k/(a+b+c+k) which proves that such triangles must be obtuse when k>0 and acute when k<0. When k=0, the triangles are Pythagorean, as in the Beiler reference and Ron Knott’s link.
%C For all k, the area of a Pythagorean k-triangle ABC with a<=b<c equals sqrt((2ab)^2-(k(a+b-c))^2))/4.
%C Define a Pythagorean k-triple <a,b,c> to be primitive if and only if there are no integers r>1, s>0 such that <a,b,c> = <rd,re,rf> for some Pythagorean s-triple <d,e,f>. Thus, every Pythagorean 1-triple is primitive. For every k>1, the set of Pythagorean k-triples contains some non-primitive triples.
%C In particular, for d a proper divisor of k, it includes (k/d)*(a,b,c), where (a,b,c) is a Pythagorean d-triple. - Franklin T. Adams-Watters, Dec 01 2011
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%e 2*3 + 2*3 = 3*4
%e 3*4 + 5*6 = 6*7
%e 4*5 + 9*10 = 10*11
%e 5*6 + 6*7 = 8*9
%e 5*6 + 14*15 = 15*16
%e 6*7 + 9*10 = 11*12
%o (True BASIC)
%o input k
%o for a = (abs(k)-k+4)/2 to 40
%o for b = a to (a^2+abs(k)*a+2)/2
%o let t = a*(a+k)+b*(b+k)
%o let c =int((-k+ (k^2+4*t)^.5)/2)
%o if c*(c+k)=t then print a;b;c,
%o next b
%o print
%o next a
%o end
%Y Cf. A103606, A198454-A198469.
%K nonn
%O 1,1
%A _Charlie Marion_, Oct 25 2011
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