In this paper we present another characterization of (±1)-invariant sequences. We also introduce truncated Fibonacci and Lucas sequences of the second kind and show that a sequence x ∈ R∞ is (−1)-invariant(1-invariant resp.) if and only if D[ 0 x ] is perpendicular to every truncated Fibonacci(truncated Lucas resp.) sequence of the second kind where D = diag((−1), (−1), (−1), . . .).

The density of primes dividing at least one term of the Lucas sequence defined by L0(P) = 2, L1(P)= P and Ln(P) = PLn-1(P) + Ln-2(P) for n ~ 2, with P an arbitrary integer, is determined.

Recently the second author [3] posed many conjectures on monotonicity of sequences of the type ( n+1 √ an+1/ n √ an)n>N with (an)n>1 a familiar combinatorial sequence of positive integers. Throughout this paper, we set N = {0, 1, 2, . . .} and Z = {1, 2, 3, . . .}. Let A and B be integers with ∆ = A − 4B 6= 0. The Lucas sequence un = un(A,B) (n ∈ N) is defined as follows: u0 = 0, u1 = 1, and un...

In this paper, we prove several identities involving linear combinations of convolutions the generalized Fibonacci and Lucas sequences. Our results apply more generally to broader classes second-order linearly recurrent sequences with constant coefficients. As a consequence, obtain as special cases many relating exactly four amongst Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal, Jacobsthal–Luc...

We review the well-known relation between Lucas sequences and exponentiation. This leads to the observation that certain public-key cryptosystems that are based on the use of Lucas sequences have some elementary properties their re-inventors were apparently not aware of. In particular, we present a chosen-message forgery for ‘LUC’ (cf. [21; 25]), and we show that ‘LUCELG’ and ‘LUCDIF’ (cf. [22,...

T HIS provocative remark has been attributed to the famous industrial statistician, Cuthbert Daniel, by Box et al. (2005) in their well-known text on the design of experiments. Split-Plot experiments were invented by Fisher (1925) and their importance in industrial experimentation has been long recognized (Yates (1936)). It is also well known that many industrial experiments are fielded as spli...

The Fibonacci sequence {Fn} is defined by the recurrence relation Fn = Fn−1+ Fn−2, for n ≥ 2 with F0 = 0 and F1 = 1. The Lucas sequence {Ln} , considered as a companion to Fibonacci sequence, is defined recursively by Ln = Ln−1 + Ln−2, for n ≥ 2 with L0 = 2 and L1 = 1. It is well known that F−n = (−1)Fn and L−n = (−1)Ln, for every n ∈ Z. For more detailed information see [9],[10]. This paper pr...

Let Γn and Λn be the n-dimensional Fibonacci cube and Lucas cube, respectively. The domination number γ of Fibonacci cubes and Lucas cubes is studied. In particular it is proved that γ(Λn) is bounded below by ⌈ Ln−2n n−3 ⌉ , where Ln is the n-th Lucas number. The 2-packing number ρ of these cubes is also studied. It is proved that ρ(Γn) is bounded below by 2 blg nc 2 −1 and the exact values of ...