23 pages, 2 figures. This version closely corresponds the version published in "Experimental Mathematics", with minor misprint and rearrangement corrections
Let $f_1=1,f_2=2$ and $f_i=f_{i-1}+f_{i-2}$ for $i>2$ be the sequence of Fibonacci numbers. Let $Φ_h(n)$ be the quantity of partitions of natural number $n$ into $h$ different Fibonacci numbers. In terms of Zeckendorf partition of $n$ I deduce a formula for the function $Φ(n;t):=\sum_{h\geq 1}Φ_h(n)t^h$, and use it to analyze the functions $F(n):=Φ(n;1)$ and $χ(n):=Φ(n;-1)$. I obtain the least upper bound for $F(n)$ when $f_i-1\<n\<f_{i+1}-1$. It implies that $F(n)\<\sqrt{n+1}$ for any natural $n$.
I prove also that $|χ(n)|\<1$, and $\mathop{\lim}\limits_{N\to\infty}\frac{1}{N} \left(χ^2(1)+χ^2(2)\dptχ^2(N)\right)=0$.
For any $k\>2$, I define a special finite set $\mathbb{G}(k)$ of solutions of the equation $F(n)=k$, all solutions can be easily obtained from $\mathbb{G}(k)$. This construction uses a representation of rational numbers as certain continued fractions and provides with a canonical identification $\coprod_{k\>2}\mathbb{G}(k)=\G_+$, where $\G_+$ is the monoid freely generated by the positive rational numbers $<1$.
Let $Ψ(k)$ be the cardinality of $\mathbb{G}(k)$. I prove that, for $i\>2k$ and $k\>2$, the interval $[f_i-1,f_{i+1}-1]$ contains exactly $2Ψ(k)$ solutions of the equation $F(n)=k$ and offer a formula for the Dirichlet generating function of the sequence $Ψ(k)$. I formulate conjectures on the set of minimal solutions of the equations $F(n)=k$ as $k$ varies and pose some questions concerning such solutions.
We prove an exact formula for OEIS A000119, which counts partitions into distinct Fibonacci numbers. We also establish an exact formula for its mean value, and determine the asymptotic behaviour.
For many meta-Fibonacci sequences it is possible to identify a partition of the sequence into successive intervals (sometimes called blocks) with the property that the sequence behaves "similarly" in each block. This partition provides insights into the sequence properties. To date, for any given sequence, only ad hoc methods have been available to identify this partition. We apply a new concept - the spot-based generation sequence - to derive a general methodology for identifying this partition for a large class of meta-Fibonacci sequences. This class includes the Conolly and Conway sequences…
In this study, we define a new type of Fibonacci and Lucas num- bers which are called bicomplex Fibonacci and bicomplex Lucas numbers. We obtain the well-known properties e.g. Docagnes, Cassini, Catalan for these new types. We also give the identities of negabicomplex Fibonacci and negabi- complex Lucas numbers, Binet formulas and relations of them.
In this paper, we define a variant of Fibonacci-like sequences that we call prime Fibonacci sequences, where one takes the sum of the previous two terms and returns the smallest odd prime divisor of that sum as the next term. We prove that these sequences always terminate in a power of two but can be extended infinitely to the left.
In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer sequence is closely related to the function counting binary partitions of a certain type.
In this paper, we introduce the bi-periodic Lucas matrix sequence and present some fundamental properties of this generalized matrix sequence. Moreover, we investigate the important relationships between the bi-periodic Fibonacci and Lucas matrix sequences. We express that some behaviours of bi-periodic Lucas numbers also can be obtained by considering properties of this new matrix sequence. Finally, we say that the matrix sequences as Lucas, $k$-Lucas and Pell-Lucas are special cases of this generalized matrix sequence.
The Fibonacci numbers are the prototypical example of a recursive sequence, but grow too quickly to enumerate sets of integer partitions. The same is true for the other classical sequences $a(n)$ defined by Fibonacci-like recursions: the tribonacci, Padovan, Pell, Narayana's cows, and Lucas sequences. For each sequence $a(n)$, however, we can define a related sequence $\textrm{sa}(n)$ by defining $\textrm{sa}(n)$ to have the same recurrence and initial conditions as $a(n)$, except that $\textrm{sa}(2n)=\textrm{sa}(n)$. Growth is no longer a problem: for each $n$ we construct recursively a set …
In this note, infinite series involving Fibonacci and Lucas numbers are derived by employing formulae similar to that which Roger Apéry utilized in his seminal paper proving the irrationality of $ζ(3)$.
In this note, we obtain some identities for the generalized Fibonacci polynomial by using the Q(x) matrix. These identities including the Cassini identity and Honsberger formula can be applied to some polynomial sequences, such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials, Fermat polynomials, Fermat-Lucas polynomials, and so on.
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of partitions into odd parts and the generating function for the number of odd divisors.
It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the strong divisibility property. However, this property does not hold for every second order sequence. In this paper we study the generalized Fibonacci polynomials and classify them in two types depending on their Binet formula. We give a complete characterization for those polynomials that satisfy the strong divisibility property. We also giveformulas to calculate the $\gcd$ of those polynomials that do not satisfy the strong divisibility property. Note. This paper is now published…
In this paper we consider the sequence of Kakutani's $α$-refinements corresponding to the inverse of golden ratio (which we call Kakutani-Fibonacci sequence of partitions) and associate to it an ergodic interval exchange (which we call Kakutani-Fibonacci transformation) using the "cutting-stacking" technique. We prove that the orbit of the origin under this map coincides with a low discrepancy sequence (which we call Kakutani-Fibonacci sequence of points), which has been also considered by other authors.
In 2007, Tachiya gave necessary and sufficient conditions for the transcendence of certain infinite products involving Fibonacci numbers $F_k$ and Lucas numbers $L_k$. In the present note, we explicitly evaluate two classes of his algebraic examples. Special cases are$$\prod_{n=1}^{\infty}(1+\frac{1}{F_{2^n+1}})=\frac{3}ϕ, \qquad \prod_{n=1}^{\infty}(1+\frac{1}{L_{2^n+1}})=3-ϕ\, ,$$where $ϕ=(1+\sqrt{5})/2$ is the golden ratio.
In this paper, we compute the spectral norms of the matrices related with integer squences and we give some example related with Fibonacci, Lucas, Pell and Perrin numbers.
A special case of an elegant result due to Anderson proves that the number of $(s,s+1)$-core partitions is finite and is given by the Catalan number $C_s$. Amdeberhan recently conjectured that the number of $(s,s+1)$-core partitions into distinct parts equals the Fibonacci number $F_{s+1}$. We prove this conjecture by enumerating, more generally, $(s,ds-1)$-core partitions into distinct parts. We do this by relating them to certain tuples of nested twin-free sets. As a by-product of our results, we obtain a bijection between partitions into distinct parts and partitions into odd parts, which p…
In this article we obtain an explicit formula in terms of the partitions of the positive integer $n$ to express the $n$-th term of a wide class of sequences of numbers defined by recursion. Our proof is based only on arithmetics. We compare our result with similar formulas obtained with different approaches already in the XIX century. Examples are given for Bernoulli, Euler and Fibonacci numbers.
In these notes we study the $k$-generalized Fibonacci sequences - $(F_n^{(k)})_{n\in \Z}$ - with positive and negative indices. Denote $T_k(x)$ its characteristic polynomial. Our most interesting finding is that if $k$ is even then the absolute value of the second real root of $T_k(x)$ is minimal among the roots. Combining this with a deep result of Bugeaud and Kaneko \cite{BK} we prove that there are only finitely many perfect powers in $(F_n^{(k)})_{n\in \Z}$, provided $k$ is even. Another consequence is that, if $k$ and $l$ denote even integers then the equation $F_m^{(k)} = \pm F_n^{(l)}$ …
In this note, we present a simple summation formula for $k$-bonacci numbers. The derivation consists in obtaining the generating function of such numbers, and noting that its evaluation at a particular value yields a formula generalizing a known expression for Fibonacci numbers.
Let $f: \mathbb{Z}_+\rightarrow \mathbb{Z}_+$ be a polynomial with the property that corresponding to every prime $p$ there exists an integer $\ell$ such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed results between the number of partitions of an integer $n$ whose parts are taken from the sequence $\{f(\ell)\}_{\ell=1}^{\infty}$ and the number of parts of those partitions which are in a certain arithmetic progression.
Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive terms of the Lucas sequence, although such partitions need not be unique. In this paper, we prove that a natural number can have at most two distinct non-consecutive partitions in the Lucas sequence, find all positive integers with a fixed term in their partition, and calculate the limiting value of the proportion of natural numbers that are not uniquely partitioned into the sum of non-consecutive te…
Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. Let $p_\mathcal{A}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the sequence $(p_\mathcal{A}(n,k) \pmod{m})_{n\in\mathbb{N}}$ for an arbitrary fixed integer $m\geqslant2$. We investigate periodicity of the sequence and lower and upper bounds for the density of the set $\{n\in\mathbb{N}: p_\mathcal{A}(n,k) \equiv i \pmod{m}\}$ for a fixed positive integer $k$ and $i\in\{0,1,\ldots, m-1\}$. In particular, we apply our results t…
Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. We study the difference between the number of summands in the partition of two consecutive integers. In particular, let $L(n)$ be the number of summands in the partition of $n$. We characterize all positive integers such that $L(n) > L(n+1)$, $L(n) < L(n+1)$, and $L(n) = L(n+1)$. Furthermore, we call $n+1$ a peak of $L$ if $L(n) < L(n+1) > L(n+2)$ and a divot of $L$ if $L(n) > L(n+1) < L(n+2)$. We characterize all such peaks and divots of $L$.
Nicolas and DeSalvo and Pak proved that the partition function $p(n)$ is log concave for $n \geq 25$. Chen, Jia and Wang proved that $p(n)$ satisfies the third order Turán inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for $n \geq 94$. More recently, Griffin, Ono, Rolen and Zagier proved more generally that for all $d$, the degree $d$ Jensen polynomials associated to $p(n)$ are hyperbolic for sufficiently large $n$. In this paper, we prove that the same result holds for the $k$-regular partition function $p_k(n)$ for $k \geq 2$. In particular, for any positive i…
Let $t_n = (-1)^{s_2(n)}$, where $s_2(n)$ is the sum of binary digits function. The sequence $(t_n)_{n\in \mathbb N}$ is the well-known Prouhet-Thue-Morse sequence. In this note we initiate the study of the sequence $(h_n)_{n\in \mathbb N}$, where $h_0 = 0, h_1 = $1 and for $n \ge 2$ we define $h_n$ recursively as follows:$ h_n = t_n h_{n-1} + h_{n-2}$. We prove several results concerning arithmetic properties of the sequence $(h_n )_{n\in \mathbb N}$. In particular, we prove non-vanishing of $h_n$ for $n \ge 5$, automaticity of the sequence $(h_n \pmod m)_{n\in \mathbb N}$ for each m, and oth…
In this note, we will give proofs of two congruences involving broken 3-diamond partitions and broken 5-diamond partitions which were conjectured by Peter Paule and Silviu Radu.
Let $Σ_{2n}$ be the set of all partitions of the even integers from the interval $(4,2n], n>2,$ into two odd prime parts. We show that $\midΣ_{2n}\mid\sim 2n^2/\log^2{n}$ as $n\to\infty$. We also assume that a partition is selected uniformly at random from the set $Σ_{2n}$. Let $2X_n\in (4,2n]$ be the size of this partition. We prove a limit theorem which establishes that $X_n/n$ converges weakly to the maximum of two random variables which are independent copies of a uniformly distributed random variable in the interval $(0,1)$. Our method of proof is based on a classical Tauberian theorem du…
In this note we present the solution of Problem H-691 (The Fibonacci Quarterly, 50 (1) 2012) with some corrections and more details. The solution involves three nontrivial integrals whose evaluations are given here.
