A198465 - OEIS

%I #8 Jul 07 2016 23:48:50

%S 3,3,4,4,6,7,5,10,11,6,7,9,6,15,16,7,10,12,7,21,22,8,28,29,9,11,14,9,

%T 36,37,10,14,17,10,22,24,10,45,46,11,27,29,11,55,56,12,15,19,12,21,24,

%U 12,66,67,13,18,22,13,25,28,13,78,79,14,45,47,14,91,92

%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).  See A198453 for more about Pythagorean k-triples.

%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 3*2 + 3*2 = 4*3

%e 4*3 + 6*5 = 7*6

%e 5*4 + 10*9 = 11*10

%e 6*5 + 7*6 = 9*8

%e 6*5 + 15*14 = 16*15

%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, A198453-A198469.

%K nonn

%O 1,1

%A _Charlie Marion_, Dec 19 2011