Leonardo number

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][4]

A Leonardo prime is a Leonardo number that is 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)

Modulo cycles

The Leonardo numbers form a cycle in any modulo n≥2. An easy way to see it is:

• 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.

The cycles for n≤8 are:

Modulo	Cycle	Length
2	1	1
3	1,1,0,2,0,0,1,2	8
4	1,1,3	3
5	1,1,3,0,4,0,0,1,2,4,2,2,0,3,4,3,3,2,1,4	20
6	1,1,3,5,3,3,1,5	8
7	1,1,3,5,2,1,4,6,4,4,2,0,3,4,1,6	16
8	1,1,3,5,1,7	6

The cycle always end on the pair (1,n-1), as it's the only pair which can precede the pair (1,1).

Expressions

• The following equation applies:

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

Proof

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

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

Leonardo polynomials

The Leonardo polynomials ${\displaystyle L_{n}(x)}$ is defined by ^[5]

${\displaystyle L_{n+2}(x)=xL_{n+1}(x)+L_{n}(x)+x}$ with ${\displaystyle L_{0}(x)=1,L_{1}(x)=2x-1.}$

Equivalently, in homogeneous form, the Leonardo polynomials can be writtenas

${\displaystyle L_{n+3}(x)=(x+1)L_{n+2}(x)-(x-1)L_{n+1}(x)-L_{n}(x):}$

where ${\displaystyle L_{0}(x)=1,L_{1}(x)=2x-1}$ and ${\displaystyle L_{2}(x)=2x^{2}+1.}$

Examples of Leonardo polynomials

${\displaystyle L_{0}(x)=1\,}$

${\displaystyle L_{1}(x)=2x-1\,}$

${\displaystyle L_{2}(x)=2x^{2}+1\,}$

${\displaystyle L_{3}(x)=2x^{3}+4x-1\,}$

${\displaystyle L_{4}(x)=2x^{4}+6x^{2}+1\,}$

${\displaystyle L_{5}(x)=2x^{5}+8x^{3}+6x-1\,}$

${\displaystyle L_{6}(x)=2x^{6}+10x^{4}+12x^{2}+1\,}$

${\displaystyle L_{7}(x)=2x^{7}+12x^{5}+20x^{3}+8x-1\,}$

${\displaystyle L_{8}(x)=2x^{8}+14x^{6}+30x^{4}+20x^{2}+1\,}$

${\displaystyle L_{9}(x)=2x^{9}+16x^{7}+42x^{5}+40x^{3}+10x-1\,}$

${\displaystyle L_{10}(x)=2x^{10}+18x^{8}+56x^{6}+70x^{4}+30x^{2}+1\,}$

${\displaystyle L_{11}(x)=2x^{11}+20x^{9}+72x^{7}+112x^{5}+70x^{3}+12x-1\,}$

Substituting ${\displaystyle x=1}$ in the above polynomials gives the Leonardo numbers and setting ${\displaystyle x=k}$ gives the k-Loenardo numbers ^[6].