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A198464
Consider triples a<=b<c where (a^2+b^2-c^2)/(c-a-b) =3, ordered by a and then b; sequence gives c values.
0
4, 8, 13, 19, 9, 13, 26, 17, 34, 43, 14, 18, 53, 32, 64, 38, 76, 19, 23, 30, 89, 103, 59, 118, 24, 28, 42, 67, 134, 151, 35, 43, 169, 29, 33, 63, 94, 188, 104, 208, 47, 58, 229, 31, 34, 38, 84, 251, 56, 137, 274, 149, 298, 39, 43, 108, 323, 52, 71, 88, 349
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 Knottā€™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.
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
KEYWORD
nonn
AUTHOR
Charlie Marion, Nov 26 2011
STATUS
approved