OFFSET

1,1

COMMENTS

We can prove that for every positive integer n there exists a triple (x,y,z) of positive integers such that x^2 + n*x*y + y^2 = z^2. One of the solutions is (s^2 - t^2, n*t^2 + 2*s*t, s^2 + n*s*t + t^2).

LINKS

Mathematics StackExchange, Finding all pairs (a,b) of positive integers such that a^2+nab+b^2 is a perfect square.

EXAMPLE

a(6)=7 because there exists a triple (2,3,7) satisfying 2^2 + 6*2*3 + 3^2 = 7^2, and no smaller solution exists.

MATHEMATICA

Table[First[

Sort@Flatten[

Table[Sqrt[x^2 + n*x*y + y^2], {x, 1, 50}, {y, x + 1, 50}]],

IntegerQ[#] &], {n, 1, 30}]

PROG

(Python)

def a(n):

for z in range(1, 100):

for y in range(z, 1, -1):

for x in range(1, y):

if x**2+n*x*y+y**2==z**2:

return(z)

print([a(n) for n in range(1, 30)])

CROSSREFS

KEYWORD

nonn

AUTHOR

Zhining Yang, Mar 11 2023

EXTENSIONS

More terms from Sean A. Irvine, Apr 10 2023

STATUS

approved