

A198455


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


2



2, 5, 9, 6, 14, 9, 20, 27, 10, 35, 13, 21, 44, 26, 54, 14, 20, 65, 17, 24, 77, 44, 90, 14, 18, 33, 51, 104, 21, 38, 119, 135, 22, 49, 75, 152, 25, 55, 84, 170, 35, 45, 189, 26, 39, 50, 68, 209, 29, 35, 75, 114, 230, 125
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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 Knottās link.
For all k, the area of a Pythagorean ktriangle ABC with a<=b<c equals sqrt((2ab)^2(k(a+bc))^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. 104134.


LINKS

Table of n, a(n) for n=1..54.
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, A156681, A198454A198469, A333530, A333531.
Sequence in context: A065225 A175640 A204913 * A339174 A018878 A021389
Adjacent sequences: A198452 A198453 A198454 * A198456 A198457 A198458


KEYWORD

nonn


AUTHOR

Charlie Marion, Oct 26 2011


STATUS

approved



