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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.
(history; published version)

#10 by N. J. A. Sloane at Thu Jul 07 23:48:49 EDT 2016

LINKS

Ron Knott, <a href="http://www.mcsmaths.surrey.ac.uk/Personalhosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>

Discussion

Thu Jul 07

23:48

OEIS Server: https://oeis.org/edit/global/2532

#9 by R. J. Mathar at Thu Feb 28 03:55:37 EST 2013

STATUS

editing

approved

#8 by R. J. Mathar at Thu Feb 28 03:55:33 EST 2013

CROSSREFS

Cf. A103606, A198454-A198472A198469.

STATUS

approved

editing

#7 by Russ Cox at Fri Mar 30 17:23:21 EDT 2012

AUTHOR

_Charlie Marion (charliemath(AT)optonline.net), _, Nov 15 2011

Discussion

Fri Mar 30

17:23

OEIS Server: https://oeis.org/edit/global/129

#6 by T. D. Noe at Fri Nov 18 12:21:08 EST 2011

STATUS

proposed

approved

#5 by Charlie Marion at Fri Nov 18 08:25:09 EST 2011

STATUS

editing

proposed

#4 by Charlie Marion at Fri Nov 18 08:25:04 EST 2011

DATA

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

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

LINKS

Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>

STATUS

proposed

editing

#3 by Charlie Marion at Tue Nov 15 19:20:03 EST 2011

STATUS

editing

proposed

Discussion

Wed Nov 16

13:38

T. D. Noe: See A198458.

#2 by Charlie Marion at Tue Nov 15 19:19:43 EST 2011

NAME

allocated for Charlie MarionConsider 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.

DATA

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

Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>

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-A198472.

KEYWORD

allocated

nonn,new

AUTHOR

Charlie Marion (charliemath(AT)optonline.net), Nov 15 2011

STATUS

approved

editing

#1 by Charlie Marion at Tue Oct 25 10:07:08 EDT 2011

NAME

allocated for Charlie Marion

KEYWORD

allocated

STATUS

approved