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TheLeonardo numbers are a sequence of numbers given by the recurrence:

Edsger W. Dijkstra^[1] used them as an integral part of hissmoothsortalgorithm,^[2] and also analyzed them in some detail.^[3][4]

ALeonardo prime is aLeonardo number that is alsoprime.

The term "Leonardo number" was coined by Dikjstra^{[5][nb 1]}, and the derivation is not given explicitly. Given the close relationship to the famous sequence credited toLeonardo Fibonacci, he may have considered the subject trivial. There is no known nor likely connection toLeonardo da Vinci, the most common subject of thatmononym.

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, ... (sequenceA001595 in theOEIS)

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, ... (sequenceA145912 in theOEIS)

The Leonardo numbers form a cycle in any modulo${\displaystyle n\ge 2}$. An easy way to see it is:

• If a pair of numbers modulo${\displaystyle n}$ appears twice in the sequence, then there is a cycle.
• If we assume the main statement is false, using the previous statement, then it would imply there are an infinite number of distinct pairs of numbers between${\displaystyle 0}$ and${\displaystyle n-1}$, which is false since there are${\displaystyle n^2}$ such pairs.

The cycles for${\displaystyle n\le 8}$ are:

The cycle always end on the pair${\displaystyle (1,n-1)}$, as it is the only pair which can precede the pair${\displaystyle (1,1)}$.

• The following equation applies:

${\displaystyle L(n)=2L(n-1)-L(n-3)}$

The Leonardo numbers are related to theFibonacci numbers by the relation${\displaystyle L(n)=2F(n+1)-1,n\ge 0}$.

From this relation it is straightforward to derive aclosed-form expression for the Leonardo numbers, analogous to Binet's formula for the Fibonacci numbers:

${\displaystyle L(n)=2\frac{{\phi }^{n+1}-{\psi }^{n+1}}{\phi -\psi }-1=\frac{2}{\sqrt{5}}\left({\phi }^{n+1}-{\psi }^{n+1}\right)-1=2F(n+1)-1}$

where thegolden ratio${\displaystyle \phi =\left(1+\sqrt{5}\right)/2}$ and${\displaystyle \psi =\left(1-\sqrt{5}\right)/2}$ are the roots of thequadratic polynomial${\displaystyle x^2-x-1=0}$.

The Leonardo polynomials${\displaystyle L_n(x)}$ is defined by^[6]

${\displaystyle L_{n+2}(x)=x{L}_{n+1}(x)+L_n(x)+x}$ 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${\displaystyle L_2(x)=2x^2+1.}$

${\displaystyle L_0(x)=1}$

${\displaystyle L_1(x)=2x-1}$

${\displaystyle L_2(x)=2x^2+1}$

${\displaystyle L_3(x)=2x^3+4x-1}$

${\displaystyle L_4(x)=2x^4+6x^2+1}$

${\displaystyle L_5(x)=2x^5+8x^3+6x-1}$

${\displaystyle L_6(x)=2x^6+10x^4+12x^2+1}$

${\displaystyle L_7(x)=2x^7+12x^5+20x^3+8x-1}$

${\displaystyle L_8(x)=2x^8+14x^6+30x^4+20x^2+1}$

${\displaystyle L_9(x)=2x^9+16x^7+42x^5+40x^3+10x-1}$

${\displaystyle L_{10}(x)=2x^{10}+18x^8+56x^6+70x^4+30x^2+1}$

${\displaystyle L_{11}(x)=2x^{11}+20x^9+72x^7+112x^5+70x^3+12x-1}$

Substituting${\displaystyle x=1}$ in the above polynomials gives the Leonardo numbers and setting${\displaystyle x=k}$ gives the${\displaystyle k}$-Leonardo numbers.^[7]

1. P. Catarino, A. Borges (2019): On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89(1), 75-86. Retrieved fromhttp://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

• OEISsequence A001595

• Integer sequences
• Fibonacci numbers
• Recurrence relations

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