Leonardo number

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The Leonardo numbers are a sequence of numbers given by the recurrence:

${\displaystyle L(n)={\begin{cases}1&{\mbox{if }}n=0\\1&{\mbox{if }}n=1\\L(n-1)+L(n-2)+1&{\mbox{if }}n>1\\\end{cases}}}$

Edsger W. Dijkstra^[1] used them as an integral part of his smoothsort algorithm,^[2] and also analyzed them in some detail.^[3]

A Leonardo prime is a Leonardo number that's also prime.

Values[]

The first few Leonardo numbers are

1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, ... (sequence A001595 in the OEIS)

The first few Leonardo primes are

3, 5, 41, 67, 109, 1973, 5167, 2692537, 11405773, 126491971, 331160281, 535828591, 279167724889, 145446920496281, 28944668049352441, 5760134388741632239, 63880869269980199809, 167242286979696845953, 597222253637954133837103, ... (sequence A145912 in the OEIS)

Expressions[]

• The following equation applies:

${\displaystyle L(n)=2L(n-1)-L(n-3)}$

Relation to Fibonacci numbers[]

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 ${\displaystyle x^{2}-x-1=0}$.

References[]

External links[]

• OEIS sequence A001595