In this paper, we define and investigate the generalized Friedrich sequences deal with, in detail, two special cases, namely, Friedrich-Lucas sequences. We present Binet's formulas, generating functions, Simson summation formulas for these Moreover, give some identities matrices related with Furthermore, show that there are close relationsbetween Friedrich, third order Jacobsthal, modified thir...

Let E be a real quadratic field with discriminant d ≡ 0 (mod p) where p is an odd prime. In terms of a Lucas quotient, the fundamental unit and the class number of E, we determine 0<c<d, (d c)=ρ p−1 ⌊pc/d⌋ modulo p 2 where ρ = ±1.

The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac–Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subs...

We prove that the number of 1-factorisations of a generalised Petersen graph of the type GP (3k, k) is equal to the kth Jacobsthal number J(k) if k is odd, and equal to 4J(k), when k is even. Moreover, we verify the list colouring conjecture for GP (3k, k).

Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. allowing zero as some Ouvry and Polychronakos 2019. We give bijection between these two varieties of determine various combinatorial properties multicompositions. In particular, we the count multicompositions number all parts, positive zeros. Then, working from three types r...

In this work, we define higher-order Jacobsthal–Lucas quaternions with the help of numbers. We examine some identities quaternions. introduce their basic definitions and properties. give Binet’s formula, Cassini’s identity, Catalan’s d’Ocagne generating functions, exponential functions also relations between Jacobsthal