A198457 - OEIS

%I #33 Oct 21 2019 14:04:04

%S 3,6,7,4,4,6,5,16,17,6,10,12,7,8,11,7,30,31,8,18,20,9,14,17,9,48,49,

%T 10,12,16,10,28,30,11,70,71,12,18,22,12,40,42,13,16,21,13,30,33,13,96,

%U 97,14,25,29,14,54,56,15,22,27,15,40,43,15,126,127,16,20,26

%N Consider triples (a, b, c) where a <= b < c and (a^2+b^2-c^2)/(c-a-b) = 2, 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. For all k, the area of a Pythagorean k-triangle ABC with a <= b < c equals sqrt((2*a*b)^2 - (k*(a+b-c))^2))/4.

%C Because either all sides or only one side of a Pythagorean (-+2)-triangle ABC is even their sum is always even. Thus csc(C) = -(a+b+c+k)/k is an integer. So ((a+2)^2 + (b+2)^2 - (c+2)^2)|(2*(a+2)*(b+2)) resp. (a^2 + b^2 - c^2)|(2*a*b). - _Ralf Steiner_, Sep 18 2019

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

%H David A. Corneth, <a href="/A198457/b198457.txt">Table of n, a(n) for n = 1..9999</a>

%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-A198468.

%K nonn

%O 1,1

%A _Charlie Marion_, Nov 09 2011

%E More terms from _David A. Corneth_, Sep 22 2019