How to find the recurrence relationship from an equation?

Question

I have the restriction $8n^2-8n+1 = m^2$ and I want to find the recurrence relationship of all integers n that satisfy this equation. I have figured out a recurrence relationship of $A_n$ = $6A_{n-1}$ - $A_{n-2}$ + 2, but I don't know how to prove that this is correct.

Can anyone explain it to me? Thanks a lot!

What does satisfying the equation mean? Do you want some recurring sequence $a_n=f(a_{n-1})$ such that $8a_n^2-8a_n+1=m^2$,... but what is $m$ supposed to be? $m=a_{n+1}$? — FShrike, Aug 08 '21 at 13:24
Or is $m$ an arbitrary integer? And you want all $a_n$ to satisfy the equation, defining $a_0$ and $f$ in terms of $m$? — FShrike, Aug 08 '21 at 13:25

J. W. Tanner · Answer 1 · 2021-08-08T19:44:30.873

1

Your equation can be written as a negative Pell equation $m^2-2(2n-1)^2=-1$, which satisfies the well-known recurrence $2a_n-1=6(2a_{n-1}-1)-(2a_{n-2}-1)$, which simplifies to $a_n=6a_{n-1}-a_{n-2}-2$.

(The sequence of solutions for $2n-1$ in the above equation is given in OEIS. )

edited Aug 08 '21 at 19:44

answered Aug 08 '21 at 19:27

J. W. Tanner

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How to find the recurrence relationship from an equation?

1 Answers1