1. Some combinatorics
Every aditive expression of a natural number n as
an ordered sequence of natural numbers is called a
decomposition [18]
of n. All the decompositions of n we denote by n'.
The number of decompositions of n is d'(n) = 2^{n1}.
We also distinguish a subclass of decompositions, denoted by n'',
where every decomposition is identified with its obverse. The number of
n''decompositions is
If n is decomposed in k numbers, the number
of such kdecompositions of n is


Every decomposition of n by numbers 2 or 1 is
called a bicomposition of n. The number of bicompositions
of n, denoted by ^{~}n, is ^{~}b(n)
= f_{n}, where f_{n} is Fibonacci sequence,
determined by the recursion formula
For the subclass of bicompositions n, where
every bicomposition is identified with its obverse, their number b(n)
is given by the recursion formula
Every ^{~}n or nbicomposition
with the number 2 occuring lk times and 1 occuring k times
is called ^{~}(l,k) or (l,k)bicomposition,
and l is called the lenght of bicomposition. There are
(l,k)bicompositions,
and among them s(l,k) are
symmetrical, where s(l,k)=
Also, we will consider the
subclasses of n' and
n''decompositions, ^{~} n and nbicompositions,
beginning with any number except 1, denoted respectively by n ^{*},
n ^{**}, ^{~}n
^{*} ,
n ^{*}. Their numbers are:
