Leonardo Number

The Leonardo numbers are a sequence of numbers given by the recurrence:
:$L(n) = \begin 1 & \mbox n = 0 \\ 1 & \mbox n = 1 \\ L(n - 1) + L(n - 2) + 1 & \mbox n > 1 \\ \end$

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 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, ...

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:
* 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:

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:
:$L(n)=2L(n-1)-L(n-3)$

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:
:$L(n) = 2 \frac- 1 = \frac \left(\varphi^ - \psi^\right) - 1 = 2F(n+1) - 1$

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 $x^2 - x - 1 = 0$.

Leonardo polynomials

The Leonardo polynomials $L_(x)$ is defined by
:$L_(x)=xL_(x)+L_(x)+x$ with $L_(x) = 1, L_(x)=2x-1.$

Equivalently, in homogeneous form, the Leonardo polynomials can be writtenas
:$L_(x)= (x+1)L_(x)-(x-1)L_(x)-L_(x):$
where $L_(x)=1, L_(x)=2x-1$ and $L_(x)=2x^2+1.$

Examples of Leonardo polynomials

:$L_(x)=1 \,$
:$L_(x)=2x-1\,$
:$L_(x)=2x^2+1\,$
:$L_(x)=2x^3+4x-1\,$
:$L_(x)=2x^4 + 6x^2 + 1 \,$
:$L_(x)=2x^5+8x^3+6x-1 \,$
:$L_(x)=2x^6+10x^4+12x^2+1\,$
:$L_(x)=2x^7+12x^5+20x^3+8x-1 \,$
:$L_(x)=2 x^8+14 x^6+30 x^4+20 x^2+1 \,$
:$L_(x)=2 x^9+16 x^7+42 x^5 + 40x^3 + 10x-1 \,$
:$L_(x)=2 x^+18 x^8+56 x^6+70 x^4+30 x^2+1\,$
:$L_(x)=2 x^+20 x^9+72 x^7+112 x^5+70 x^3+12x-1 \,$

Substituting $x=1$ in the above polynomials gives the Leonardo numbers and setting $x=k$ 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

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

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{{Classes of natural numbers

Integer sequences
Fibonacci numbers
Recurrence relations