Leonardo number

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

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 used them as an integral part of his smoothsort algorithm, and also analyzed them in some detail.

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, ...

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, ...

Modulo cycles
The Leonardo numbers form a cycle in any modulo n≥2. An easy way to see it is: The cycles for n≤8 are: The cycle always end on the pair (1,n-1), as it's the only pair which can precede the pair (1,1).
 * If a pair of numbers modulo n appears twice in the sequence, then there's a cycle.
 * If we assume the main statement is false, using the previous statement, then it would imply there's infinite distinct pairs of numbers between 0 and n-1, which is false since there are n2 such pairs.

Expressions

 * The following equation applies:
 * $$L(n)=2L(n-1)-L(n-3)$$

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

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

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