# Finding all squares in a generalized Fibonacci type sequence

I asked this two weeks ago at https://math.stackexchange.com/questions/3571697/squares-in-a-second-order-integer-recursive-sequence

Most of the related question shown to me while composing are about primes, not squares...

Given a sequence $$x_n$$ as in https://oeis.org/A001075 $$1, 2, 7, 26, 97, 362, 1351,$$ such that $$x_{n+2} = 4 x_{n+1} - x_n$$

These are the $$x$$ values in $$x^2-3y^2 = 1$$

Can we find, and prove, all squares in the sequence and all double squares? I see that Cohn did this for the Fibonacci and Lucas numbers in the 1960's. For this sequence, it seems $$1$$ is the only square and $$2$$ is the only doubled square.

alright, an answer at the recent Squares in Lucas sequences does give a reference Petho

• Is this sequence somehow special? (and why?) – Wlod AA Mar 24 at 9:38