Leonardo number

The first few Leonardo numbers are

${\displaystyle 1,\;1,\;3,\;5,\;9,\;15,\;25,\;41,\;67,\;109,\;177,\;287,\;465,\;753,\;1219,\;1973,\;3193,\;5167,\;8361,\ldots }$ (sequence A001595 in the OEIS)

The Leonardo numbers are related to the Fibonacci numbers by the relation ${\displaystyle L(n)=2F(n+1)-1,n\geq 0}$.

From this relation it is straightforward to derive a closed-form expression for the Leonardo numbers, analogous to Binet's formula for the Fibonacci numbers:

${\displaystyle L(n)=2{\frac {\varphi ^{n+1}-\psi ^{n+1}}{\varphi -\psi }}-1={\frac {2}{\sqrt {5}}}\left(\varphi ^{n+1}-\psi ^{n+1}\right)-1=2F(n+1)-1}$

where the golden ratio ${\displaystyle \varphi =\left(1+{\sqrt {5}}\right)/2}$ and ${\displaystyle \psi =\left(1-{\sqrt {5}}\right)/2}$ are the roots of the quadratic polynomial ${\displaystyle x^{2}-x-1=0}$.

• OEIS sequence A001595