125 research outputs found
The Binet formula, sums and representations of generalized Fibonacci p-numbers
AbstractIn this paper, we consider the generalized Fibonacci p-numbers and then we give the generalized Binet formula, sums, combinatorial representations and generating function of the generalized Fibonacci p-numbers. Also, using matrix methods, we derive an explicit formula for the sums of the generalized Fibonacci p-numbers
On a generalization of the Pell sequence
The Pell sequence is the second order linear recurrence defined by with initial conditions and . In this paper, we investigate a generalization of the Pell sequence called the -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
K-Fibonacci sequences and minimal winning quota in Parsimonious game
Parsimonious games are a subset of constant sum homogeneous weighted majority
games unequivocally described by their free type representation vector. We show
that the minimal winning quota of parsimonious games satisfies a second order,
linear, homogeneous, finite difference equation with nonconstant coefficients
except for uniform games. We provide the solution of such an equation which may
be thought as the generalized version of the polynomial expansion of a proper
k-Fibonacci sequence. In addition we show that the minimal winning quota is a
symmetric function of the representation vector; exploiting this property it is
straightforward to prove that twin Parsimonious games, i.e. a couple of games
whose free type representations are each other symmetric, share the same
minimal winning quota
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