A198457 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.

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, 10, 12, 16, 10, 28, 30, 11, 70, 71, 12, 18, 22, 12, 40, 42, 13, 16, 21, 13, 30, 33, 13, 96, 97, 14, 25, 29, 14, 54, 56, 15, 22, 27, 15, 40, 43, 15, 126, 127, 16, 20, 26

Offset

1,1

Comments

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).

If a, b and c form a Pythagorean k-triple, then na, nb and nc form a Pythagorean nk-triple.

A triangle is defined to be a Pythagorean k-triangle if its sides form a Pythagorean k-triple.

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.

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

References

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

Links

David A. Corneth, Table of n, a(n) for n = 1..9999

Ron Knott, Pythagorean Triples and Online Calculators

Example

3*5 +  6*8  =  7*9;
4*6 +  4*6  =  6*8;
5*7 + 16*17 = 17*18;
6*8 + 10*12 = 12*14;
7*9 +  8*10 = 11*13;
7*9 + 30*32 = 31*33.

Prog

(True BASIC)
input k
for a = (abs(k)-k+4)/2 to 40
   for b = a to (a^2+abs(k)*a+2)/2
      let t = a*(a+k)+b*(b+k)
      let c = int((-k+(k^2+4*t)^.5)/2)
      if c*(c+k)=t then print a; b; c,
   next b
   print
next a
end

Crossrefs

Cf. A103606, A198454-A198468.

Sequence in context: A177035 A055102 A243976 * A264921 A098990 A336453

Adjacent sequences: A198454 A198455 A198456 * A198458 A198459 A198460

Keyword

nonn

Author

Charlie Marion, Nov 09 2011

Extensions

More terms from David A. Corneth, Sep 22 2019

Status

approved