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A277269 Hypotenuses of Pythagorean triples, generated by a variation of Euclid's formula.

%I A277269 #29 May 08 2019 15:49:42
%S A277269 5,10,13,17,10,25,26,29,34,41,37,20,15,26,61,50,53,58,65,74,85,65,34,
%T A277269 73,20,89,50,113,82,85,30,97,106,39,130,145,101,52,109,58,25,68,149,
%U A277269 82,181,122,125,130,137,146,157,170,185,202,221,145,74,51,40,169,30,75,122,265,170,173,178,185,194,205,218,233,250,269,290,313,197,100,205,106,221,116,35,130,277,148,317,170,365,226,229
%N A277269 Hypotenuses of Pythagorean triples, generated by a variation of Euclid's formula.
%C A277269 Take two positive integers, x > y. As shown in the referenced faux art, you can form a vector using the integers as the coordinates, and repeat that vector and its equal-length normal so that you get exactly to the x-axis. Now you can mirror the pattern: take the same number of normal vectors but in the opposite direction. You get an isosceles triangle and the equal sides represent a Pythagorean triple.
%C A277269 Let s = gcd(x,y). This is the scaling factor -- you divide x and y by it and get coprime x and y. The symmetry axis goes from (0,0) to (xx,xy). The first normal goes from (xx,xy) to (xx+yy,0). The second normal goes from (xx,xy) to (xx-yy,xy+xy). So x^2+y^2 is the hypotenuse of the triangle with catheti x^2-y^2 and 2xy. Scale these with s and you get the triple corresponding to the parameters. In the examples the hypotenuse will be called P(x,y).
%H A277269 Juhani Heino, <a href="/A277269/a277269.png">Faux art</a> showing the motivation of this.
%H A277269 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean Triple</a>.
%e A277269 Triangle with each row r going from P(r+1,1) to P(r+1,r):
%e A277269 P(2,1)=5;
%e A277269 P(3,1)=10, P(3,2)=13;
%e A277269 P(4,1)=17, P(4,2)=2*P(2,1)=10, P(4,3)=25;
%e A277269 P(5,1)=26, P(5,2)=29, P(5,3)=34, P(5,4)=41;
%e A277269 P(6,1)=37, P(6,2)=2*P(3,1)=20, P(6,3)=3*P(2,1)=15, P(6,4)=2*P(3,2)=26, P(6,5)=61;
%o A277269 (PARI)
%o A277269 p(x,y) = x^2 + y^2
%o A277269 out=""
%o A277269 for (row = 1, 15, for (col = 1, row, s=gcd(row+1, col); out = Str(out, s * p((row+1)/s, col/s),", ") ))
%o A277269 print(out);
%Y A277269 When results are ordered and doubles removed, we should get A009003.
%Y A277269 A222946 is similar but omits non-primitive triples (gives 0 for them).
%K A277269 nonn,tabl,easy
%O A277269 1,1
%A A277269 _Juhani Heino_, Oct 16 2016