A198459 - OEIS

%I #10 Jul 07 2016 23:48:49

%S 6,4,16,10,8,30,18,14,48,12,28,70,18,40,16,30,96,25,54,22,40,126,20,

%T 33,70,160,26,42,88,24,64,198,52,108,30,78,240,28,40,63,130,54,286,34,

%U 48,75,154,32,64,110,336,88,180,38,128,390,28,36,66,102,208,448,33,42

%N Consider triples a<=b<c where (a^2+b^2-c^2)/(c-a-b) =2, ordered by a and then b; sequence gives b values.

%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. 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.

%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*5 + 6*8 = 7*9

%e 4*6 + 4*6 = 6*8

%e 5*7 + 16*17 = 17*18

%e 6*8 + 10*12 12*14

%e 7*9 + 8*10 = 11*13

%e 7*9 + 30*32 = 31*33

%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_, Nov 15 2011