[seqfan] Re: How to list pairs, triples, etc.

[Richard Guy]

Well, that went up like a lead balloon!  No-one noticed
the triples  {159,130,66}  {160,129,96}  {182,107,14},
{190,134,35}  and  {192,132,64}  showing that I missed
159, 160, 182, 190 and 192 from the sequence.  How many
others did I miss?  I can usually rely on my mistakes to
attract attention.

The general formula for antipythagorean triples is easily
seen to be

(y^2 + z^2 - x^2)/2, (z^2 + x^2 - y^2)/2, (x^2 + y^2 - z^2)/2

and if the triples are to comprise distinct positive integers,
then  x, y, z  are the edges of an acute scalene triangle, an
even number of which are odd.      R.

On Sat, 15 May 2010, Richard Guy wrote:

> Dear all,
>          Perhaps this is already well known to those who
> well know it, but is there a way to list triples, say, in
> OEIS ?  Partial answer: the hypotenuses of Pythagorean
> triples are listed in A009003.
>
>         Here is a sequence which seems not to be already
> in, which someone may like to submit.  It arises from
> what might be called antipythagorean triples, namely
> {a,b,c} where  b+c = x^2,  c+a=y^2,  a+b=z^2  are each
> perfect squares.  Now any positive (or indeed negative)
> integer can occur in an antipythagorean triple (in
> fact in infinitely many, I believe), but I am interested
> in which positive integers can be the largest member of
> an antipythagorean triple.  Perhaps all sufficiently
> large numbers, but that's an open question as far as
> I'm presently concerned.
>
> Some numbers can be (the largest) members of more than
> one antipythagorean triple, e.g., {120,24,1} {120,76,24}
> or {126,99,70} {126,100,44} --- should these (largest
> members) appear more than once in the sequence?
>
> [I sh'd've said that I want my triples to consist of
> distinct positive integers, for example, I don't count
> (288,288,1} though some might think that I should.]
>
> I get the first few antipythagorean triples to be
>
> {30,19,6} {44,20,5} {47,34,2} {48,33,16} {60,21,4}
> {66,34,15} {69,52,12} {70,51,30} {78,22,3} {86,35,14}
> {90,54,10} {92,52,29} {94,75,6} {95,74,26} ...
>
> so that the proposed sequence is (E&OE, as usual)
>       [there were plenty of those!]
> 30, 44, 47, 48, 60, 66, 69, 70, 78, 86, 90, 92, 94,
> 95, 96, 98, 108, 113, 116, 118, 118, 120 (twice?),
> 122, 125, 126 (twice?), 132, 138, 142, 147, 150, 152,
> 154, 156, 157, 158 (twice?), 165, 170, 176, 180, 185,
> 186, 188, 194 (twice?), 195, 196, 197, 198, 200, 207,
> 212, 214, 216, 218, 221, 222, 224, 227, 230, 232, ...
>
> Let me know of any I've missed, and if you can prove
> that all sufficiently large numbers are in there
> (or not!) and what is meant by ``sufficiently large''.
>
> Thanks to anyone interested.   R.