The further considerations in this paper concern number sequence tasks where a finite initial segment of a linear-recursive sequence of rational numbers is given and one or more subsequent members as a ''correct'' continuation of the sequence are sought.
Many number sequence tasks in psychological applications do not demand that the subject give an explicit rule or formula but only a ''correct'', ''regular'' or ''logical'' continuation or completion of the presented number sequence. For our analysis, however, the detailed version of a number sequence task is essential. Referring to the previously definined notions, we first give a precise and explicit formulation of the general type of linear-recursive number sequence task. We presuppose that the type of rules sought are specified in the task instruction (linear recursion formulae), however, that no essential restriction is imposed on the domain of recursion formulae; that is, the domain for the recursion formulae in this general type of task is . After the formulation of the general type of linear-recursive number sequence task, we will describe the problems of existence and uniqueness of solutions for this type of number sequence task. In Section 4, these problems will be discussed in further detail.
3.1 The general type of linear-recursive number sequence task.
Given a (non-constant)
n-term sequence of rational
numbers , a maximal permissible degree
of recursion, and the number k for a k-fold
extension of .
One easily sees that for each
n-term sequence is linear-recursive of degree
because a solution for
is given, for instance, by
Generally, when some recursion formula for a given sequence is
found, the subsequent members can be calculated.
Not for each sequence , however, does there
exist an such
that is -recursive with
(Example: The sequence is not
linear-recursive of some degree r<5).
Therefore, the mathematical analysis of a number sequence task
basically involves examining the problem of solvability.
3.2 The problem of solvability.
Given a (non-constant) n-term number sequence and a maximal permissible degree of recursion. How can it be determined whether the sequence task of type 3.1 is solvable ?
In practical application the solvability of a task of type 3.1 can be guaranteed simply by constructing the number sequence task according to an appropriate recursion formula of some degree . Nevertheless, even if the solvability of a given sequence task of type 3.1 is assured and there exists some solution, there is another more profound problem with the sequence task, namely the problem of uniqueness of the solution.
3.3 The problem of uniqueness of solution.
Let be an n-term sequence for which an exists such that is -recursive and is the k-fold -extension.
The formulation of the uniqueness problem indicates fundamental problems in the application of number sequence tasks in psychology that are by no means artificial:
-recursive with the term ,
-recursive with the term ,
-recursive with the term ,
-recursive with the term .
In Sections 5 and 6 we will discuss the uniqueness problem 3.3 in connection with a special type of natural-number sequences where the domain for the permissible recursion formulae is substantially restricted. First, in Section 4 some general results concerning the existence and uniqueness problems of recursion formulae for given number sequences are derived; that is, we will concentrate on the uniqueness problems of type 3.3 I. and leave a more general treatment of problems of type 3.3 II. for another time.