In number theory, the sequence of numbers $L(n)$ defined by the initial values $L(0)=2$ and $L(1)=1,$

and the recurrence relation $L(n)=L(n-1)+L(n-2)$ for $n2$ is called the Lucas sequence.

As in the diagram at left, the sequence can be

illustrated geometrically as the side lengths of squares

arranged in a spiral, where you can see that the Lucas

sequence begins $2,1,3,4,7,11,18,29,47,76.$ The

spiral is a little wonky at the center but smooths out.

Backing around the spiral makes a geometric argument

that $L(n)=L(n-1)+L(n-3)+L(n-4).$ Many other such

relationships exist as well.

Use mathematical induction to prove this claim about a sum of the squares of Lucas numbers:

Claim: ${\displaystyle \underset{k=0}{\overset{n}{}}}{L}_k^2=L_n*L_{n+1}+2$