Related papers: Sums of powers over equally spaced Fibonacci numbe…

Powers of Fibonacci polynomials are expressed as single sums, improving on a double sum recently seen in the literature.

We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.

In this paper, we find all the sums of three Fibonacci numbers which are close to a power of 2. This paper continues and extends the previous work of Hasanalizade \cite{Hasanalizade}.

In this paper, we consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.

We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.

Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.

Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular…

We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.

We offer several new summation identities involving harmonic numbers, odd harmonic numbers, and Fibonacci numbers. Our results are derived using three different approaches: partial summation, polynomial identities and binomial…

In this paper, we find all sums of two Fibonacci numbers which are close to a power of 2. As a corollary, we also determine all Lucas numbers close to a power of 2. The main tools used in this work are lower bounds for linear forms in…

For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,\ldots,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for…

In part 1 of this paper some linear weighted generalized Fibonacci number summation identities were derived using the fact that the Fibonacci number is the residue of a rational function. In this part, using the same method, some quadratic…

Balancing numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of balancing numbers can be summed explicitly. For this, as a first step, a power $B_n^l$ is expressed as a linear combination of…

In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of…

We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.

We give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.

We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.

In this paper, we find the closed sums of certain type of Fibonacci related convergent series. In particular, we generalize some results already obtained by Brousseau, Popov, Rabinowitz and others.

We study sums of powers of Fibonacci and Lucas polynomials of the form $% \sum_{n=0}^{q}F_{tsn}^{k}(x) $ and $\sum_{n=0}^{q}L_{tsn}^{k}% (x) $, where $s,t,k$ are given natural numbers, together with the corresponding alternating sums…

We extend a result of I. J. Good and prove more symmetry properties of sums involving generalized Fibonacci numbers