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]

References

1. "E.W.Dijkstra Archive: Fibonacci numbers and Leonardo numbers. (EWD 797)". www.cs.utexas.edu. Retrieved 2020-08-11.
2. Dijkstra, Edsger W. Smoothsort – an alternative to sorting in situ (EWD-796a) (PDF). E.W. Dijkstra Archive. Center for American History, University of Texas at Austin. ( transcription )
3. "E.W.Dijkstra Archive: Smoothsort, an alternative for sorting in situ (EWD 796a)". www.cs.utexas.edu. Retrieved 2020-08-11.
4. "Leonardo Number - GeeksforGeeks". www.geeksforgeeks.org. 18 October 2017. Retrieved 2022-10-08.
5. Kalika Prasad, Munesh Kumari (2024): The Leonardo polynomials and their algebraic properties. Proc. Indian Natl. Sci. Acad. https://doi.org/10.1007/s43538-024-00348-0
6. Kalika Prasad, Munesh Kumari (2025): The generalized k-Leonardo numbers: a non-homogeneous generalization of the Fibonacci numbers, Palestine Journal of Mathematics, 14.

Cited

1. P. Catarino, A. Borges (2019): On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89(1), 75-86. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1005/799

2. K. Prasad, R. Mohanty, M. Kumari, H. Mahato (2024): Some new families of generalized k-Leonardo and Gaussian Leonardo Numbers, Communications in Combinatorics and Optimization, 9 (3), 539-553. https://comb-opt.azaruniv.ac.ir/article_14544_6844cc9ba641d31cafe5358297bc0096.pdf

3. M. Kumari, K. Prasad, H. Mahato, P. Catarino (2024): On the generalized Leonardo quaternions and associated spinors, Kragujevac Journal of Mathematics 50 (3), 425-438. https://imi.pmf.kg.ac.rs/kjm/pub/kjom503/kjm_50_3-6.pdf

4. K. Prasad, H. Mahato, M. Kumari, R. Mohanty: On the generalized Leonardo Pisano octonions, National Academy Science Letters 47, 579–585. https://link.springer.com/article/10.1007/s40009-023-01291-2

5. Y. Soykan (2023): Special cases of generalized Leonardo numbers: Modified p-Leonardo, p-Leonardo-Lucas and p-Leonardo Numbers, Earthline Journal of Mathematical Sciences. https://www.preprints.org/frontend/manuscript/a700d41e884b69f92bc8eb6cf7ff1979/download_pub

6. Y. Soykan (2021): Generalized Leonardo numbers, Journal of Progressive Research in Mathematics. https://core.ac.uk/download/pdf/483697189.pdf

7. E. Tan, HH Leung (2023): ON LEONARDO p-NUMBERS, Journal of Combinatorial Number Theory. https://math.colgate.edu/~integers/x7/x7.pdf

External links

• OEIS sequenceA001595