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Fibonacci expansion

Every positive integer can be expressed uniquely as a sum of Fibonacci numbers $F_k$ with the condition that consecutive Fibonacci numbers $F_{k+1}, F_k$ are not used. This expansion may be computed using a greedy algorithm: given $N$, subtract the largest $F_k$ with $F_k \le N$ and repeat with $N' = N - F_k$.

Tribonacci sequence

A linear recursive sequence, an extension of the sequence of Fibonacci numbers having the recurrence relation $$ t_{n+3}=t_{n+2}+t_{n+1}+t_n $$ for $n\ge 0$ and initial conditions $$ t_0=a\ ,t_1=b,\ t_2=c, $$ (with $a$, $b$, $c$ given constants).

The concept was introduced by the fourteen-year-old student M. Feinberg in 1963 in [a1] for the case: $a=b=1$, $c=2$: the term 'Tribonacci numbers is used for this specific case. The basic properties are introduced in [a5], [a3], [a4], [a2].

The formula for the $n$-th number is given by A. Shannon in [a3]: $$ T_n = \sum_{m=0}^{[n/2]} \sum_{r=0}^{[n/3]} \binom{ n-m-2r }{ m+r }\binom{ m+r }{ r } $$

Binet's formula for the $n$-th number is given by W. Spickerman in [b2]: $$ T_n = \frac{\rho^{n+2}}{ (\rho-\sigma)(\rho-\bar\sigma) } + \frac{\sigma^{n+2}}{ (\sigma-\rho)(\sigma-\bar\sigma) } + \frac{\bar\sigma^{n+2}}{ (\bar\sigma-\rho)(\bar\sigma-\sigma) } $$ where $$ \rho = \frac{1}{3}\left({ (19+3\sqrt{33})^{1/3} + (19-3\sqrt{33})^{1/3} +1 }\right)\,, $$ and $$ \sigma = \frac{1}{6}\left({ 2-(19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) + \frac{\sqrt3 i}{6}\left({ (19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) \ . $$ Here $\rho,\sigma,\bar\sigma$ are the roots of the companion polynomial $z^3 = z^2 + z + 1$.

The Tribonacci numbers count the binary sequences with no occurrences of three consecutive digits $\mathbf{1}$. [Ref?]

The Tribonacci sequence was generalized in [a6], [a7] to the form of two sequences:

$$a_{n+3}=u_{n+2}+w_{n+1}+y_n,$$

$$b_{n+3}=v_{n+2}+x_{n+1}+z_n,$$

where $u,v,w,x,y,z\in\{a,b\}$ and each of the tuples $(u,v)$, $(w,x)$, $(y,z)$ contains the two symbols $a$ and $b$. There are eight different such schemes. An open problem (as of 2000) is the construction of an explicit formula for each of them.

References

[a1]	M. Feinberg, "Fibonacci–Tribonacci" The Fibonacci Quart. , 1 : 3 (1963) pp. 71–74
[a2]	C. Valavigi, "Properties of Tribonacci numbers" The Fibonacci Quart. , 10 : 3 (1972) pp. 231–246
[a3]	A. Scott, T. Delaney, V. Hoggatt Jr., "The Tribonacci sequence" The Fibonacci Quart. , 15 : 3 (1977) pp. 193–200
[a4]	A. Shannon, "Tribonacci numbers and Pascal's pyramid" The Fibonacci Quart. , 15 : 3 (1977) pp. 268–275 Zbl 0385.05006
[a5]	I. Bruce, "A modified Tribonacci sequence" The Fibonacci Quart. , 22 : 3 (1984) pp. 244–246
[a6]	K. Atanassov, J. Hlebarova, S. Mihov, "Recurrent formulas of the generalized Fibonacci and Tribonacci sequences" The Fibonacci Quart. , 30 : 1 (1992) pp. 77–79
[a7]	J.-Z. Lee, J.-S. Lee, "Some properties of the generalization of the Fibonacci sequence" The Fibonacci Quart. , 25 : 2 (1987) pp. 111–117
[b2]	W. Spickerman, "Binet's formula for the Tribonacci sequence" The Fibonacci Quart. , 20 (1981) pp. 118–120 Zbl 0486.10011

Density of a sequence

A concept in general additive number theory, in which one studies addition laws for sequences of general form. The density of a sequence is a measure of what part of the sequence of all natural numbers belongs to a given sequence $A=\{a_k\}$ of integers $a_0=0<1\leq a_1<\ldots<a_k$. Let $$ A(x) = \sum_{a_i \le x \\ a_i \in A}\,1 $$ be the counting function of $A$.

Schnirelmann density. $$ \sigma(A) = \inf_n \frac{A(n)}{n} $$.

Let $W = \sum_n w_n$ be a divergent series of positive real numbers and let $W(x) = \sum_{n \le x} w_n$ be its summatory function. We can define a density relative to $W$ $$ d_W(X) = \lim_{x \rightarrow \infty} \frac{A(x)}{W(x)} $$ if this limit exists, and upper and lower densities to be the corresponding upper and lower limits if these exist.

Asymptotic density, a particular case of this being the natural density. This is density with respect to the "natural" series $w_n = 1$.

Logarithmic density. This is density with respect to the series $w_n = 1/n$. If a sequence has natural density then it has logarithmic density and the two are equal, but the converse does not hold.

Let $P(\sigma)$ denote a probability distribution on natural numbers, that is, a sequence $p_n = p_n(\sigma)$ such that $\p_n \ge 0$ and $\sum_n p_n$ converges to 1; we suppose that $P(\sigma)$ depends on a parameter $\sigma$.

The concept of density is also extended to numerical sequences differing from the natural sequence, for example to the sequence of integers in algebraic number fields. As a result it is possible to examine bases in algebraic fields.

References

[1]	A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian)
[2]	H.H. Ostmann, "Additive Zahlentheorie" , Springer (1956)
[3]	Gérald Tenenbaum, "Introduction to analytic and probabilistic number theory" , (tr. C.B.Thomas) Cambridge University Press (1995) ISBN 0-521-41261-7

Schnirelmann Density

A definition of the density of a sequence, introduced in 1930 by L.G. Shnirel'man.

By the density of a sequence $A$ one means the density $d(A)$ of $A$, namely

$$d(A)=\inf\frac{A(n)}{n},$$

where

$$A(n)=\sum_{1\leq a_k\leq n}1.$$

The density $d(A)=1$ if and only if $A$ coincides with the set $\mathbf N_0$ of all non-negative integers. Let $A+B$ be the arithmetic sum of two sequences $A=\{a_k\}$ and $B=\{b_t\}$, i.e. the set $A+B=\{a_k+b_t\}$ where the numbers $a_k+b_t$ are taken without repetition. If $A=B$, one puts $2A=A+A$, and similarly $3A=A+A+A$, etc. If $hA=\mathbf N_0$, then $A$ is called a basis of order $h$. On examining the structures of sets obtained by summing sequences determined only by their densities, one uses the following theorems on the density of the sum of two sequences:

$$d(A+B)\geq d(A)+d(B)-d(A)d(B)$$

(Shnirel'man's inequality) and the stronger inequality

$$d(A+B)\geq\min(d(A)+d(B),1)$$

(the Mann–Dyson inequality).

Shnirel'man's inequality implies that any sequence of positive density is a basis of finite order. This can be used in additive problems, in which one frequently sums sequences of zero density by the preliminary construction of new sequences with positive density from the given ones. For example, it has been shown by sieve methods that the sequence $\{p\}+\{p\}$, where $p$ runs through the prime numbers, has positive density. Shnirel'man's theorem follows from this: There exists an integer $c_0>0$ such that any natural number is the sum of at most $c_0$ prime numbers. This theorem gives a solution to the so-called weak Goldbach problem (see also Additive number theory).

A variant of this concept of density is that of asymptotic density, a particular case of this being the natural density. The concept of density is also extended to numerical sequences differing from the natural sequence, for example to the sequence of integers in algebraic number fields. As a result it is possible to examine bases in algebraic fields.

References

[1]	A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian)
[2]	H.H. Ostmann, "Additive Zahlentheorie" , Springer (1956)