${\displaystyle \left[x^{2}+y^{2}+(x+y)^{2}\right]^{2}=2[(x^{4}+y^{4}+(x+y)^{4}]}$ to prove for fibonacci numbers:

${\displaystyle (f_{n}^{2}+f_{n+1}^{2}+f_{n+2}^{2})^{2}=2(f_{n}^{4}+f_{n+1}^{4}+f_{n+2}^{4})}$