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{{Short description|Set of numbers used in the smoothsort algorithm}}
{{primary sources|date=July 2017}}
The '''Leonardo numbers''' are a sequence of numbers given by the recurrence:
:<math> 
  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}
 </math>

[[Edsger W. Dijkstra]]<ref>{{Cite web|title=E.W.Dijkstra Archive: Fibonacci numbers and Leonardo numbers. (EWD 797)|url=http://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD797.html|access-date=2020-08-11|website=www.cs.utexas.edu}}</ref> used them as an integral part of his [[smoothsort]] [[algorithm]],<ref>{{Cite EWD|796a|Smoothsort – an alternative to sorting in situ}}</ref>  and also analyzed them in some detail.<ref>{{Cite web|title=E.W.Dijkstra Archive: Smoothsort, an alternative for sorting in situ (EWD 796a)|url=http://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD796a.html|access-date=2020-08-11|website=www.cs.utexas.edu}}</ref><ref>{{Cite web |title=Leonardo Number - GeeksforGeeks |url=https://www.geeksforgeeks.org/leonardo-number/ |access-date=2022-10-08 |website=www.geeksforgeeks.org|date=18 October 2017}}</ref>

A '''Leonardo prime''' is a '''Leonardo number''' that is also [[Prime number|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, ... {{OEIS|id=A001595}}

The first few Leonardo primes are
:[[3]], [[5]], [[41 (number)|41]], [[67 (number)|67]], [[109 (number)|109]], 1973, 5167, 2692537, 11405773, 126491971, 331160281, 535828591, 279167724889, 145446920496281, 28944668049352441, 5760134388741632239, 63880869269980199809, 167242286979696845953, 597222253637954133837103, ... {{OEIS|id=A145912}}

==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 n<sup>2</sup> such pairs.
The cycles for n≤8 are:
{|class="wikitable"
|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:
:<math>L(n)=2L(n-1)-L(n-3)</math>
{{Math proof|title=Proof|proof=<math>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)</math>}}

==Relation to Fibonacci numbers==
The Leonardo numbers are related to the [[Fibonacci number]]s by the relation <math>L(n) = 2 F(n+1) - 1, n \ge 0</math>.

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:
:<math>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</math>

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

==Leonardo polynomials ==
The Leonardo polynomials <math> L_{n}(x) </math> is defined by <ref>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</ref>
:<math>L_{n+2}(x)=xL_{n+1}(x)+L_{n}(x)+x </math> with <math>L_{0}(x) = 1, L_{1}(x)=2x-1. </math>

Equivalently, in homogeneous form, the Leonardo polynomials can be writtenas
:<math>L_{n+3}(x)= (x+1)L_{n+2}(x)-(x-1)L_{n+1}(x)-L_{n}(x):</math>
where <math>L_{0}(x)=1, L_{1}(x)=2x-1 </math> and <math> L_{2}(x)=2x^2+1.</math>

==Examples of Leonardo polynomials  ==
:<math>L_{0}(x)=1 \,</math>
:<math>L_{1}(x)=2x-1\,</math>
:<math>L_{2}(x)=2x^2+1\,</math>
:<math>L_{3}(x)=2x^3+4x-1\,</math>
:<math>L_{4}(x)=2x^4 + 6x^2 + 1 \,</math>
:<math>L_{5}(x)=2x^5+8x^3+6x-1 \,</math>
:<math>L_{6}(x)=2x^6+10x^4+12x^2+1\,</math>
:<math>L_{7}(x)=2x^7+12x^5+20x^3+8x-1 \,</math>
:<math>L_{8}(x)=2 x^8+14 x^6+30 x^4+20 x^2+1 \,</math>
:<math>L_{9}(x)=2 x^9+16 x^7+42 x^5 + 40x^3 + 10x-1 \,</math>
:<math>L_{10}(x)=2 x^{10}+18 x^8+56 x^6+70 x^4+30 x^2+1\,</math> 
:<math>L_{11}(x)=2 x^{11}+20 x^9+72 x^7+112 x^5+70 x^3+12x-1 \,</math>

Substituting <math>x=1</math> in the above polynomials gives the Leonardo numbers and setting <math>x=k</math> gives the k-Leonardo numbers.<ref>Kalika Prasad, Munesh Kumari (2025): The generalized k-Leonardo numbers: a non-homogeneous generalization of the Fibonacci numbers, ''Palestine Journal of Mathematics, 14.''</ref>

==References==
{{reflist}}

== 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 el|sequencenumber=A001595|name=|formalname=a(n) = a(n-1) + a(n-2) + 1, with a(0) = a(1) = 1}}

{{Classes of natural numbers}}

[[Category:Integer sequences]]
[[Category:Fibonacci numbers]]
[[Category:Recurrence relations]]