ただし、sgn x は符号関数である。
行列表現

また、フィボナッチ数列の漸化式は次のように行列表現できる[3]： ( F n + 2 F n + 1 ) = ( 1 1 1 0 ) ( F n + 1 F n ) {\displaystyle {F_{n+2} \choose F_{n+1}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{n+1} \choose F_{n}}} ∴ ( F n + 1 F n F n F n − 1 ) = ( 1 1 1 0 ) n {\displaystyle \therefore {\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}}

nを2nで置換すると、 ( F 2 n + 1 F 2 n F 2 n F 2 n − 1 ) = ( 1 1 1 0 ) 2 n = ( ( 1 1 1 0 ) n ) 2 = ( F n + 1 F n F n F n − 1 ) 2 = ( F n 2 + F n + 1 2 ( F n − 1 + F n + 1 ) F n ( F n − 1 + F n + 1 ) F n F n − 1 2 + F n 2 ) {\displaystyle {\begin{aligned}{\begin{pmatrix}F_{2n+1}&F_{2n}\\F_{2n}&F_{2n-1}\end{pmatrix}}&={\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{2n}=\left({\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}\right)^{2}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}^{2}\\&={\begin{pmatrix}{F_{n}}^{2}+{F_{n+1}}^{2}&(F_{n-1}+F_{n+1})F_{n}\\(F_{n-1}+F_{n+1})F_{n}&{F_{n-1}}^{2}+{F_{n}}^{2}\end{pmatrix}}\end{aligned}}}

よって、 F 2 n + 1 = F n 2 + F n + 1 2 F 2 n = ( F n − 1 + F n + 1 ) F n = ( 2 F n − 1 + F n ) F n = ( 2 F n + 1 − F n ) F n F 2 n − 1 = F n − 1 2 + F n 2 {\displaystyle {\begin{array}{lcl}F_{2n+1}&=&{F_{n}}^{2}+{F_{n+1}}^{2}\\F_{2n}&=&(F_{n-1}+F_{n+1})F_{n}\\&=&(2F_{n-1}+F_{n})F_{n}\\&=&(2F_{n+1}-F_{n})F_{n}\\F_{2n-1}&=&{F_{n-1}}^{2}+{F_{n}}^{2}\end{array}}}