A198456 Consider triples a<=b<c where (a^2+b^2-c^2)/(c-a-b) =1, ordered by a and then b; sequence gives c values.

3, 6, 10, 8, 15, 11, 21, 28, 13, 36, 16, 23, 45, 28, 55, 18, 23, 66, 21, 27, 78, 46, 91, 20, 23, 36, 53, 105, 26, 41, 120, 136, 28, 52, 77, 153, 31, 58, 86, 171, 40, 49, 190, 33, 44, 54, 71, 210, 36, 41, 78, 116

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.

The definition amounts to saying that T_a+T_b=T_c where T_i denotes a triangular number (A000217). - N. J. A. Sloane, Apr 01 2020

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..52.

Ron Knott, Pythagorean Triples and Online Calculators

J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

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. A000217, A156682, A198454-A198469, A333530, A333531.

Sequence in context: A333576 A009019 A198467 * A337602 A360946 A032570

Adjacent sequences: A198453 A198454 A198455 * A198457 A198458 A198459

Keyword

nonn

Author

Charlie Marion, Oct 26 2011

Status

approved