A198453 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.

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, 36, 9, 13, 16, 9, 21, 23, 9, 44, 45, 10, 26, 28, 10, 54, 55, 11, 14, 18, 11, 20, 23, 11, 65, 66, 12, 17, 21, 12, 24, 27

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((2ab)^2-(k(a+b-c))^2))/4.

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.

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

References

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

Links

Table of n, a(n) for n=1..60.

Ron Knott, Pythagorean Triples and Online Calculators

Example

2*3 + 2*3 = 3*4
3*4 + 5*6 = 6*7
4*5 + 9*10 = 10*11
5*6 + 6*7 = 8*9
5*6 + 14*15 = 15*16
6*7 + 9*10 = 11*12

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-A198469.

Sequence in context: A277321 A262365 A063988 * A345162 A316313 A325876

Adjacent sequences: A198450 A198451 A198452 * A198454 A198455 A198456

Keyword

nonn

Author

Charlie Marion, Oct 25 2011

Status

approved