

A198453


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


4



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
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OFFSET

1,1


COMMENTS

The definition can be generalized to define Pythagorean ktriples a<=b<c where (a^2+b^2c^2)/(cab)=k, or where for some integer k, a(a+k) + b(b+k) = c(c+k).
If a, b and c form a Pythagorean ktriple, then na, nb and nc form a Pythagorean nktriple.
A triangle is defined to be a Pythagorean ktriangle if its sides form a Pythagorean ktriple.
If a, b and c are the sides of a Pythagorean ktriangle 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 ktriangle ABC with a<=b<c equals sqrt((2ab)^2(k(a+bc))^2))/4.
Define a Pythagorean ktriple <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 striple <d,e,f>. Thus, every Pythagorean 1triple is primitive. For every k>1, the set of Pythagorean ktriples contains some nonprimitive triples.
In particular, for d a proper divisor of k, it includes (k/d)*(a,b,c), where (a,b,c) is a Pythagorean dtriple.  Franklin T. AdamsWatters, Dec 01 2011


REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104134.


LINKS



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



KEYWORD

nonn


AUTHOR



STATUS

approved



