cp's OEIS Frontend

A positive integer n is in this sequence if and only if there is a solution to the Pell-like equation x^2-ny^2=d^2(n-1)n(n+1)/3 for some x,y,d integers.

A positive integer n is in this sequence if and only if it can be written in the form: (u^2-3w^2)/(v^2+3w^2), with u,v,w integers and gcd(v,w)=1. This can also be written as a n(v^2) + 3(n+1)(w^2) = z^2.

If n is in this sequence, then we can find an arithmetic progression of *positive* integers which satisfy this equation. (The description above does not require the sequence to be positive.)

By using the method of Legendre to find whether there exists rational numbers r,s on the curve nr^2 + 3(n+1)s^2 = 1, we get the following necessary and sufficient conditions on n:

A. Factor n=a^2b, with b squarefree, then

1. If 3 does not divide b(n+1), then b ≅ 1 (mod 3)

2. If b is divisible by 3, then b ≅ 6 (mod 9)

3. 3 is a square (mod b.)

B.

1. If n+1 is divisible by 3, then (n+1)/3 is the sum of two perfect squares

2. If n+1 is not divisible by 3, then n+1 is the sum of two perfect squares

When n is a perfect square, we can use the arithmetic sequence starting at m=(3n+2)(sqrt(n)-1)/2 + 6 and common difference 6.

This is A001032 with the addition of 25, because in this sequence the perfect squares may include 0. Subsequence of A134419.