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Let $a$ , $b$ and $c$ be fixed complex numbers. Let $M_{n} (a, b, c)$ be the $n \times n$ Toeplitz matrix all of whose entries above the diagonal are $a$ , all of whose entries below the diagonal are $b$ , and all of whose entries on the diagonal are $c$ . For $1 \leq k \leq n$ , each $k \times k$ principal minor of $M_{n} (a, b, c)$ has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of $M_{n} (a, b, c)$ . We also show that all complex polynomials in $M_{n} (a, b, c)$ are Toeplitz matrices. In particular, the inverse of $M_{n} (a, b, c)$ is a Toeplitz matrix when...

Non-possibility to prove infinity of regular primes form some theorems.

Ladislav Skula (1977)

Journal für die reine und angewandte Mathematik

Note on a Shannon's theorem concerning the Fibonacci numbers and Diophantine quadruples

Morgado, José (1991)

Portugaliae mathematica

Note on Legendre numbers.

Sitaramachandrarao, R., Davis, B.R. (1988)

International Journal of Mathematics and Mathematical Sciences

Note on the Chebyshev polynomials and applications to the Fibonacci numbers.

Morgado, José (1995)

Portugaliae Mathematica

Note on the local growth of iterated polynomials.

John W. Layman, Bruce Landman (1984)

Aequationes mathematicae

Observations sur les triangles rectangles en nombres entiers et les suites de Fibonacci

C.-A. Laisant (1919)

Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale

On a binary recurrent sequence of polynomials

Reinhardt Euler, Luis H. Gallardo, Florian Luca (2014)

Communications in Mathematics

In this paper, we study the properties of the sequence of polynomials given by $g_{0} = 0, g_{1} = 1$ , $g_{n + 1} = g_{n} + Δ g_{n - 1}$ for $n \geq 1$ , where $Δ \in 𝔽_{q} [t]$ is non-constant and the characteristic of $𝔽_{q}$ is $2$ . This complements some results from R. Euler, L.H. Gallardo: On explicit formulae and linear recurrent sequences, Acta Math. Univ. Comenianae, 80 (2011) 213-219.

On a constant of Turán and Erdös

D. Lehmer (1980)

Acta Arithmetica

On a Diophantine problem of Bennett

Yang Hai, P. G. Walsh (2010)

Acta Arithmetica

On a generalization of the Lucas functions

H. Williams (1972)

Acta Arithmetica

On a generalization of the Pell sequence

Jhon J. Bravo, Jose L. Herrera, Florian Luca (2021)

Mathematica Bohemica

The Pell sequence ${(P_{n})}_{n = 0}^{\infty}$ is the second order linear recurrence defined by $P_{n} = 2 P_{n - 1} + P_{n - 2}$ with initial conditions $P_{0} = 0$ and $P_{1} = 1$ . In this paper, we investigate a generalization of the Pell sequence called the $k$ -generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers are also deduced....

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Netted matrices.

New Ramanujan-type formulas and quasi-Fibonacci numbers of order 7.

Nombres de Bell et analyse $p$ -adique

Nombres de Hurwitz et unités elliptiques. Un critère de régularité pour les extensions abéliennes d'un corps quadratique imaginaire

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Non-possibility to prove infinity of regular primes form some theorems.

Note on a Shannon's theorem concerning the Fibonacci numbers and Diophantine quadruples

Note on Legendre numbers.

Note on the Chebyshev polynomials and applications to the Fibonacci numbers.

Note on the local growth of iterated polynomials.

Observations sur les triangles rectangles en nombres entiers et les suites de Fibonacci

On a binary recurrent sequence of polynomials

On a constant of Turán and Erdös

On a Diophantine problem of Bennett

On a generalization of the Lucas functions

On a generalization of the Pell sequence

On a Problem of D'Atri and Nickerson.

On a Problem of W. Sierpinski.

On a product of sines

On an asymptotic formula for the Niven numbers.

Currently displaying 201 – 220 of 444