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A198460 Consider triples a<=b<c where (a^2+b^2-c^2)/(c-a-b) =2, ordered by a and then b; sequence gives c values. 0
7, 6, 17, 12, 11, 31, 20, 17, 49, 16, 30, 71, 22, 42, 21, 33, 97, 29, 56, 27, 43, 127, 26, 37, 72, 161, 32, 46, 90, 31, 67, 199, 56, 110, 37, 81, 241, 36, 46, 67, 132, 59, 287, 42, 54, 79, 156, 41, 69, 113, 337, 92, 182, 47, 131, 391, 40, 46, 72, 106, 210, 449, 45, 52 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..64.

Ron Knott, Pythagorean Triples and Online Calculators

EXAMPLE

3*5 + 6*8 = 7*9

4*6 + 4*6 = 6*8

5*7 + 16*17 = 17*18

6*8 + 10*12 12*14

7*9 + 8*10 = 11*13

7*9 + 30*32 = 31*33

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: A259168 A078323 A099255 * A215334 A298377 A299244

Adjacent sequences:  A198457 A198458 A198459 * A198461 A198462 A198463

KEYWORD

nonn

AUTHOR

Charlie Marion, Nov 15 2011

STATUS

approved

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Last modified December 17 04:31 EST 2018. Contains 318192 sequences. (Running on oeis4.)