A198461 Consider triples a<=b<c where (a^2+b^2-c^2)/(c-a-b) = 3, ordered by a and then b; sequence gives a, b and c values in that order.

2, 3, 4, 3, 7, 8, 4, 12, 13, 5, 18, 19, 6, 6, 9, 6, 11, 13, 6, 25, 26, 7, 15, 17, 7, 33, 34, 8, 42, 43, 9, 10, 14, 9, 15, 18, 9, 52, 53, 10, 30, 32, 10, 63, 64, 11, 36, 38, 11, 75, 76, 12, 14, 19, 12, 19, 23, 12, 27, 30, 12, 88, 89, 13, 102, 103, 14, 57, 59, 14, 117, 118

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.

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..72.

Ron Knott, Pythagorean Triples and Online Calculators

Example

2*5 + 3*6 = 4*7
3*6 + 7*10 = 8*11
4*7 +12*15 = 13*16
5*8 + 18*21 = 19*22
6*9 = 6*9 = 9*12
6*9 = 11*14 = 13*16

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: A058267 A048259 A324150 * A228576 A329465 A324195

Adjacent sequences: A198458 A198459 A198460 * A198462 A198463 A198464

Keyword

nonn

Author

Charlie Marion, Nov 26 2011

Status

approved