Leonardo number

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}}

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).

• The following equation applies:

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

{{Math proof|title=Proof|proof=$L(n)=L(n-1)+L(n-2)+1=L(n-1)+L(n-2)+1+L(n-3)-L(n-3)=2L(n-1)-L(n-3)$}}

The Leonardo numbers are related to the Fibonacci numbers by the relation $L(n)\; =\; 2\; F(n+1)\; -\; 1,\; n\; \backslash ge\; 0$.

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:

:$L(n)\; =\; 2\; \backslash frac\{\backslash varphi^\{n+1\}\; -\; \backslash psi^\{n+1\}\}\{\backslash varphi\; -\; \backslash psi\}-\; 1\; =\; \backslash frac\{2\}\{\backslash sqrt\; 5\}\; \backslash left(\backslash varphi^\{n+1\}\; -\; \backslash psi^\{n+1\}\backslash right)\; -\; 1\; =\; 2F(n+1)\; -\; 1$

where the golden ratio $\backslash varphi\; =\; \backslash left(1\; +\; \backslash sqrt\; 5\backslash right)/2$ and $\backslash psi\; =\; \backslash left(1\; -\; \backslash sqrt\; 5\backslash right)/2$ are the roots of the quadratic polynomial $x^2\; -\; x\; -\; 1\; =\; 0$.

The Leonardo polynomials $L\_\{n\}(x)$ 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

:$L\_\{n+2\}(x)=xL\_\{n+1\}(x)+L\_\{n\}(x)+x$ 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 $L\_\{2\}(x)=2x^2+1.$

:$L\_\{0\}(x)=1\; \backslash ,$

:$L\_\{1\}(x)=2x-1\backslash ,$

:$L\_\{2\}(x)=2x^2+1\backslash ,$

:$L\_\{3\}(x)=2x^3+4x-1\backslash ,$

:$L\_\{4\}(x)=2x^4\; +\; 6x^2\; +\; 1\; \backslash ,$

:$L\_\{5\}(x)=2x^5+8x^3+6x-1\; \backslash ,$

:$L\_\{6\}(x)=2x^6+10x^4+12x^2+1\backslash ,$

:$L\_\{7\}(x)=2x^7+12x^5+20x^3+8x-1\; \backslash ,$

:$L\_\{8\}(x)=2\; x^8+14\; x^6+30\; x^4+20\; x^2+1\; \backslash ,$

:$L\_\{9\}(x)=2\; x^9+16\; x^7+42\; x^5\; +\; 40x^3\; +\; 10x-1\; \backslash ,$

:$L\_\{10\}(x)=2\; x^\{10\}+18\; x^8+56\; x^6+70\; x^4+30\; x^2+1\backslash ,$

:$L\_\{11\}(x)=2\; x^\{11\}+20\; x^9+72\; x^7+112\; x^5+70\; x^3+12x-1\; \backslash ,$

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.

{{reflist}}

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

• {{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