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

5, 10, 13, 17, 10, 25, 26, 29, 34, 41, 37, 20, 15, 26, 61, 50, 53, 58, 65, 74, 85, 65, 34, 73, 20, 89, 50, 113, 82, 85, 30, 97, 106, 39, 130, 145, 101, 52, 109, 58, 25, 68, 149, 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

Offset

1,1

Comments

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.

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).

Links

Table of n, a(n) for n=1..91.

Juhani Heino, Faux art showing the motivation of this.

Wikipedia, Pythagorean Triple.

Example

Triangle with each row r going from P(r+1,1) to P(r+1,r):
P(2,1)=5;
P(3,1)=10, P(3,2)=13;
P(4,1)=17, P(4,2)=2*P(2,1)=10, P(4,3)=25;
P(5,1)=26, P(5,2)=29, P(5,3)=34, P(5,4)=41;
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;

Prog

(PARI)
p(x, y) = x^2 + y^2
out=""
for (row = 1, 15, for (col = 1, row, s=gcd(row+1, col); out = Str(out, s * p((row+1)/s, col/s), ", ") ))
print(out);

Crossrefs

When results are ordered and doubles removed, we should get A009003.

A222946 is similar but omits non-primitive triples (gives 0 for them).

Sequence in context: A313384 A050127 A313385 * A185450 A072284 A242898

Adjacent sequences: A277266 A277267 A277268 * A277270 A277271 A277272

Keyword

nonn,tabl,easy

Author

Juhani Heino, Oct 16 2016

Status

approved