Leonardo number

Name

The term "Leonardo number" was coined by Dikjstra, and the derivation is not given explicitly. Given the close relationship to the famous sequence credited to Leonardo Fibonacci, he may have considered the subject trivial. There is no known nor likely connection to Leonardo da Vinci, the most common subject of that mononym.

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{\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 \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_{n}(x) is defined by :L_{n+2}(x)=xL_{n+1}(x)+L_{n}(x)+x with L_{0}(x) = 1, L_{1}(x)=2x-1.

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

Examples of Leonardo polynomials

:L_{0}(x)=1 , :L_{1}(x)=2x-1, :L_{2}(x)=2x^2+1, :L_{3}(x)=2x^3+4x-1, :L_{4}(x)=2x^4 + 6x^2 + 1 , :L_{5}(x)=2x^5+8x^3+6x-1 , :L_{6}(x)=2x^6+10x^4+12x^2+1, :L_{7}(x)=2x^7+12x^5+20x^3+8x-1 , :L_{8}(x)=2 x^8+14 x^6+30 x^4+20 x^2+1 , :L_{9}(x)=2 x^9+16 x^7+42 x^5 + 40x^3 + 10x-1 , :L_{10}(x)=2 x^{10}+18 x^8+56 x^6+70 x^4+30 x^2+1, :L_{11}(x)=2 x^{11}+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.

Notes

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

References

1. "E.W.Dijkstra Archive: Fibonacci numbers and Leonardo numbers. (EWD 797)".
2. {{Cite EWD. 796a. Smoothsort – an alternative to sorting in situ
3. "E.W.Dijkstra Archive: Smoothsort, an alternative for sorting in situ (EWD 796a)".
4. (18 October 2017). "Leonardo Number - GeeksforGeeks".
5. "E.W.Dijkstra Archive: Fibonacci numbers and Leonardo numbers. (EWD 797)".
6. 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
7. Kalika Prasad, Munesh Kumari (2025): The generalized k-Leonardo numbers: a non-homogeneous generalization of the Fibonacci numbers, ''Palestine Journal of Mathematics, 14.''