Squares of the Fibonacci numbers

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The squares of the Fibonacci numbers are, you guessed it, the Fibonacci numbers squared. Given a Fibonacci number $\scriptstyle F_{n}\,$, observe that

$F_{n-1}F_{n+1}\,=\,{F_{n}}^{2}+(-1)^{n}.\,$

For example, $\scriptstyle F_{4}\,F_{6}\,=\,3\times 8\,=\,5^{2}+(-1)^{5}\,=\,(F_{5})^{2}-1\,=\,24,\,$ and $\scriptstyle F_{5}\,F_{7}\,=\,5\times 13\,=\,8^{2}+(-1)^{6}\,=\,(F_{6})^{2}+1\,=\,65.\,$

Sequences

A007598 Squared Fibonacci numbers: $\scriptstyle F(n)^{2}\,$ where $\scriptstyle F(n)\,=\,\,$ A000045$\scriptstyle (n),\,n\,\geq \,0.\,$

{0, 1, 1, 4, 9, 25, 64, 169, 441, 1156, 3025, 7921, 20736, 54289, 142129, 372100, 974169, 2550409, 6677056, 17480761, 45765225, 119814916, 313679521, 821223649, 2149991424, ...}

See also

• Fibonacci multiplication table
• Lucas numbers
• Squares of the Lucas numbers