Leonardo number
{{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:
:
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{{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}} used them as an integral part of his smoothsort algorithm,{{Cite EWD|796a|Smoothsort – an alternative to sorting in situ}} and also analyzed them in some detail.{{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}}{{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}}
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, ... {{OEIS|id=A001595}}
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, ... {{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 n2 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 proof|title=Proof|proof=}}
Relation to Fibonacci numbers
The Leonardo numbers are related to the Fibonacci numbers by the relation .
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:
:
where the golden ratio and are the roots of the quadratic polynomial .
Leonardo polynomials
The Leonardo polynomials is defined by 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
: with
Equivalently, in homogeneous form, the Leonardo polynomials can be writtenas
:
where and
Examples of Leonardo polynomials
:
:
:
:
:
:
:
:
:
:
:
:
Substituting in the above polynomials gives the Leonardo numbers and setting gives the k-Leonardo numbers.Kalika Prasad, Munesh Kumari (2025): The generalized k-Leonardo numbers: a non-homogeneous generalization of the Fibonacci numbers, Palestine Journal of Mathematics, 14.
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}}