Klaus Korossy: Linear-Recursive Number Sequence Tasks 44 to be a prime example of inductive reasoning. Inductive reasoning has been an integral part o

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1 Methods of Psychological Research Online 1998, Vol.3, No.1 Internet: Solvability and Uniqueness of Linear-Recursive Number Sequence Tasks æ Klaus Korossy Psychologisches Institut der Universitíat Heidelberg Abstract This article deals with the problems of existence and uniqueness of solution in connection with number sequence tasks èoften called "number-series completion tasks"è that are widely used in various psychological contexts. It is argued that the problems of existence and uniqueness of solution can be rationally analyzed and controlled only when each type of sequence task applied is explicitly referred to some domain of permissible rules accounting for the included sequence type. Following this guideline, the class of linear-recursive number sequence tasks is introduced where the sequence members are related by linear recursive equations. The investigation of this type of sequence task can substantially proæt from the theory of linear equation systems. The analysis carried out uncovers several types of non-uniqueness of solution. Most striking is the fact that worst cases of non-uniqueness may occur even in strongly restricted subtypes of linear-recursive number sequences as is shown for a speciæc type of task. The results of the analyses suggest that ènumberè sequence tasks should not be applied in psychological contexts if not accompanied by an instruction that refers explicitily to some domain of permitted sequence rules. Key words: number series number-series completion tasks linear-recursive number sequences existence and uniqueness of solution 1 Number sequence tasks in psychometrics and cognitive psychology Number sequence tasks 1 are among the most frequently used numerical items in psychometric tests of intelligence or ability. In general, a sequence of about four to ten natural èless frequently rationalè numbers is presented, which either contains some discrepant element that has to be detected and marked, or which has to be continued by one or two "correct" subsequent elements. Solving number sequence tasks requires the ability to discover a general rule or relation among speciæc elements and to apply it to new elements. Similar to analogy or matrix problems, and very like letter sequence tasks, solving number sequence tasks is considered æ An earlier version of this paper was part of a report for the research project "Wissensstrukturen" that was supported by the Deutsche Forschungsgemeinschaft under Grant Lu 385è1-2 to Prof. Dr. Josef Lukas and Prof. Dr. Dietrich Albert at the University of Heidelberg. The author is grateful to the anonymous expert referees for their helpful comments on a preliminary version of this paper. 1 In psychology widely used terms are "number series problems", "number-series completion problems" etc. In this paper we prefer the term "number sequence" to the term "number series" because in mathematical terminology a "series" is deæned as a sum of terms of a sequence.

2 Klaus Korossy: Linear-Recursive Number Sequence Tasks 44 to be a prime example of inductive reasoning. Inductive reasoning has been an integral part of standardized intelligence and ability tests for many decades and is generally assumed to be a central component of many cognitive activities èholyoak, 1984; Pellegrino & Glaser, 1980è. Apart from their common use in psychometric tests, number sequence tasks as measures of inductive-reasoning skills are also frequently applied in various æelds of psychological research èfor examples see Grabitz & Wittmann, 1986; Hackett, Betz, O'Halloran & Romac, 1990; Linde & Bergstríom, 1992, experiment 2è. It seems that items utilizing letter or number sequences provide appropriate instruments both within psychometrics for measuring components of intelligence and ability as well as for investigating certain theoretical and practical problems in cognitive and experimental psychology. One reason for this suitability of sequence tasks may be the fact that number sequence tasks as well as letter series tasks allow for creating a universe of items with almost arbitrary variations in diæculty levels. In particular, number sequence tasks oæer the additional advantage that only knowledge about natural èsometimes rationalè numbers and some elementary arithmetic operations is required. Nevertheless, despite their obvious advantages there is an often voiced criticism directed to the careless use of sequence tasks in psychological contexts. Let us assume for the moment that the problem of the existence of a solution èthe solvability problemè for some number sequence task has been settled since the number sequence is constructed according to a certain rule and is thus also solvable according to this rule. There remains, however, the criticism that, generally, aside from the implemented rule of the sequence and the keyed answer considered correct, many diæering answers basing on other unintended èsay accidentalè regularities of the number sequence seem possible and might be judged to be correct. That is, even when the sequence task is solvable according to the implemented constructional rule, the solution may not be unique. This èpresupposedè problem is referred to in this paper as the ènon-èuniqueness problem of number sequence tasks. The non-uniqueness problem of number sequence tasks, however, is usually not taken seriously. Jensen è1980è, for example, rejects the criticism as "utterly trivial", "... because the other correct solutions are usually possible only for a mathematician; they involve a level of mathematical sophistication far beyond that required for the most obvious solution. Even an expert mathematician who could ægure out other possible solutions would not do so in a test situation, because it would take so much more time, and anyone capable of æguring out one of the more complex solutions would certainly have no diæculty arriving at the simplest solution, which in every item is the keyed answer" èp.153è. Particularly in the "psychometric approach" as characterized by Mayer, Larkin & Kadane è1984è, there seems to be widespread carelessness about the non-uniqueness problem of number sequence items. Possibly, a test constructor primarily engaged in developing a new test of "inductive reasoning" will be inclined to eliminate the non-uniqueness problem by identifying inappropriate èunreliable or invalidè items on the basis of empirical data and excluding such items from the test. However, to dismiss the non-uniqueness problem by purely empirical means and arguments seems not only uneconomical but from the perspective of the "cognitive approach" èmayer et al., 1984è also hardly acceptable. Cognitively oriented research on inductive reasoning has been focused on identifying the cognitive processes involved in psychometric tasks such as analogy problems èholzman, Pellegrino & Glaser, 1982; Sternberg, 1977è and series-completion problems èholzman, Pellegrino & Glaser, 1983; Kotovsky & Simon, 1973; Simon &

3 Klaus Korossy: Linear-Recursive Number Sequence Tasks 45 Kotovsky, 1963è. Prominent models of sequence completion solution, on which a large part of research has been based, include four component processes: detection of relations, discovery of periodicity, completion of pattern description, and extrapolation, where the ærst three processes combine to generate a pattern description. Conclusions about performance on series-completion problems are derived mainly from a presumed relationship between pattern descriptions and working memory: more complex pattern descriptions make greater demands on working memory. Clearly, on this level of model building, the problem of non-unique solvability of sequential items may not be ignored. In any case, modeling the solution process for some type of task has to be founded on a careful task analysis where all characteristics of the speciæc type of task and particularly all possible alternative solution ways have to be taken into account. Unfortunately, it just happens to be the class of number sequence tasks which seems so favorable in many respects, that creates considerable diæculties for the analysis of the solvability and uniqueness problems. In this paper, the solvability and ènon-èuniqueness problems will be investigated for a speciæc class of number sequence tasks. The line of argument in this investigation is that the two problems of solvability and ènon-èuniqueness of solutions can be handled eæciently only if task instruction, constructional principles of sequences and keyed answers for the applied type of number sequence tasks are explicitly referred to a limited class of permissible rules for the sequences. In fact, the considerations in this paper arose from speciæc research motivated by Doignon & Falmagne's theory of knowledge spaces èdoignon & Falmagne, 1985; see also Falmagne, Koppen, Villano, Doignon & Johannesen, 1990è. This theory has developed powerful methods for the qualitative modeling and diagnosis of knowledge based on "surmised" solution dependencies among items and corresponding "knowledge structures". However, in Doignon & Falmagne's behavioral approach the preferred methods of establishing surmise structures include èundoubtedly highly sophisticatedè expert querying procedures that are not designed to uncover the theoretical basis for the surmised solution dependencies on a set of items. In a research project entitled "Wissensstrukturen" at the University of Heidelberg èsee footnote *; for an overview see e.g. Lukas & Albert, 1993è an essential aim was to develop methods for theoretical foundation and goal-directed construction of knowledge structures. The intention was to make a contribution to reconcile Doignon & Falmagne's behavioral knowledge modeling and diagnosing approach with more traditional psychometric models and newer developments in cognitiveprocesses-approaches èsee aboveè. Two promising methods turned out to be such methods that derive hypotheses on behavioral knowledge states from the analysis of basic components of domain-speciæc tasks èe.g. Albert & Held, 1994; Held, 1993è, and methods that generate hypothesized knowledge states by utilizing appropriate task-speciæc cognitive processing models èptucha, 1994; Schrepp, 1993; 1995è. In one of these studies 2 the intention was to establish a surmise-ordered network of number sequence tasks and to test the corresponding knowledge states empirically. For the tasks a modiæed type of recursive number sequences as discussed in Krause è1985è seemed appropriate. The surmise-ordered network of number sequence tasks was constructed on the basis of a restricted set of sequence rules varying in the level of complexity. However, the unavoidable problems of solvability and uniqueness of solution for the constructed items emerged: Structuring a set of items on the basis of their solution principles necessarily requires the complete explication of all alternative ways of solving the speciæc type of item used. In the case of the type The study was carried out in an experimental practicum by Hellriegel, Ptucha &Wíolk in

4 Klaus Korossy: Linear-Recursive Number Sequence Tasks 46 of linear-recursive number sequences thematized in this paper the solvability and uniqueness problems can be treated in a relatively elementary way. The basic concept of a linear-recursive number sequence is introduced in Section 2. Section 3 presents a general type of task utilizing linear-recursive number sequences and addresses the problems of solvability and uniqueness. Section 4 contains some useful and easily derivable results on the existence, uniqueness, and structure of solution formulae for the description of linear-recursive number sequences. The central ideas of this paper are presented in Sections 5-7, where an exemplary analysis of the special type of linear-recursive number sequences applied in the previously mentioned study is conducted. Section 7 takes up the decision procedure used in Section 4 for the solution analysis of number sequence tasks and applies it to the special type of task discussed in Sections 5í7. Section 8 contains a summary and general discussion of the results of our investigation. 2 Linear-recursive number sequences: deænitions and examples Many number sequences have the characteristic property that subsequent members are related to the preceding members by linear equations. 2.1 Examples. èaè Arithmetic sequences as, for instance, the sequence 1; 3; 5; 7; 9; 11;::: can be described by giving the ærst member a 1 èhere: a 1 = 1è and the general term a i = a 1 +èi, 1èd for i =2; 3; 4; ::: èhere: d = 2è; another way of description is by giving a 1 and the recursive equation a i = a i,1 + d; i =2; 3; 4; ::: or, for instance, the initial members a 1 ;a 2 and the recursive equation a i = 2a i,1, a i,2 ; i =3; 4; 5; ::: : èbè Geometric sequences as, for instance, the sequence 24;,12; 6;,3; 3 2 ;, 3 4 ; 3 8 ; ::: can be described by giving the ærst member a 1 èhere: a 1 = 24è and the general term a i = a 1 q i,1 for i =2; 3; 4; ::: èhere: q =, 1 2è, but also recursively by giving a 1 and the equation a i = qa i,1 ; i =2; 3; 4; ::: : ècè Awell-known sequence of numbers is the Fibonacci sequence 1; 1; 2; 3; 5; 8; 13; 21;::: with the recursive description a 1 =1; a 2 =1 a i = a i,1 + a i,2 ; i =3; 4; 5; ::: : èdè The sequence 1; 3; 4; 4; 3; 1;::: is recursively described by a 1 =1; a 2 =3 a i = 2a i,1, a i,2, 1 ; i =3; 4; 5; ::: :

5 Klaus Korossy: Linear-Recursive Number Sequence Tasks 47 Number sequences like those shown in the examples above, which are widely used in psychometric tests and psychological experimentation, will be of central interest in this article. In this section we ærst introduce some basic concepts concerning the special type of number sequences shown in the examples. The following abbreviations will be used: IN the set of natural numbers f1; 2; 3; :::g èwithout 0è IQ the set of rational numbers A n := A æ ::: æ A the set of n-tupel of elements of set A íz í n,mal 2.2 Deænition. Let èa i è be a èinæniteè non-constant sequence a 1 ;a 2 ;a 3 ; ::: of rational numbers. If for a given r 2 IN there exists an èr + 1è-tupel F r := h c; f 1 ; :::; f r i2d r ç IQ r+1 with f r 6= 0 such that the recursive equations a i = rx j=1 f j a i,j + c; i = r +1;r+2; ::: è1è are satisæed, then the sequence èa i è is called linear-recursive of degree r with the recursion coeæcients f 1 ; :::; f r and the constant c, in brief h c; f 1 ; :::; f r i-recursive èor F r -recursiveè. The èr + 1è-tupel h c; f 1 ; :::; f r i is called a recursion formula èof degree rèfor èa i è, occasionally a solution to è1è; D r èwhich may be a proper subset of IQ r æèiqnf0gè è is called the domain for the recursion formulae of degree r. Remarks. èiè Within the calculus of diæerences, è1è describes an inhomogeneous linear diæerence equation with constant coeæcients and constant inhomogenity èsee e.g. Markuschewitsch, 1955; Rommelfanger, 1986, pp è, where, however, it is permitted that the recursion equation è1è may be satisæed beginning only with the members of a higher index than r + 1. Because our consideration diæers from the typical way of looking at the problems in the calculus of diæerence, we will not pursue that approach here. èiiè With respect to the types of number sequences applied in psychometrics or experimental psychology we presuppose that the members of the considered sequences, the recursion coeæcients, and the constant c are elements of the æeld IQ of rational numbers which is closed under the elementary arithmetic operations. Special attention will later be given to the domain D r ç IQ r æèiqnf0gè for the recursion formula. èiiiè In Deænition 2.2 the condition f r 6= 0 is to prevent that for instance the sequence 1; 1; 1; 1; 2; 3; :::, with a i = a i,1 +1 for i =5; 6; :::,may be classiæed as h1; 1; 0; 0; 0i-recursive. According to 2.2 this sequence is not h1; 1; 0; 0; 0i-recursive, since the condition f 4 6=0 is not satisæed. However, this sequence is also not h1; 1; i- recursive of degree 1, because the recursion equation does not hold for i =2; 3; 4 èsee Remark èièè. Instead the sequence is, for instance, h0; 1; 0; 0; 1i-recursive of degree 4, because a i = a i,1 + a i,4 holds for i =5; 6; :::. èivè It might seem reasonable to include constant number sequences in the concept of linear-recursive number sequences. We will refrain from doing so in this paper because the rather trivial and uninteresting case of a constant number sequence would demand, mainly because of notational reasons, special consideration in some proofs.

6 Klaus Korossy: Linear-Recursive Number Sequence Tasks 48 Examples. In the examples of 2.1, the sequence èaè is h2; 1i-recursive aswell as h0; 2;,1i-recursive; the sequence èbè is h0;, 1 2 i-recursive; the Fibonacci sequence ècè is h0; 1; 1i-recursive; the sequence èdè is h,1; 2;,1i-recursive. As is seen in the examples above certain sequences can be represented by several diæerent recursion formulae. On the other hand of course, each recursion formula describes inænitely many sequences depending on the given initial members. Perhaps the concept of a linear-recursive number sequence seems to be relatively speciæc. However, a large proportion of number sequences used in psychometric tests can be conceived as linear-recursive sequences although they may not appear to be so at ærst glance. The following examples may serve to verify this. 2.3 Examples. èaè The sequence 9; 7; 10; 8; 11; 9; 12;::: èi-s-t 70, Form A1, Aufgabengruppe 06, see Amthauer, 1973è is introduced in the test instruction as an example sequence accompanied by the proposed construction rule: "In this sequence alternately 2 is subtracted and 3 is added..." Nevertheless, this sequence is covered by Deænition 2.2 as a h1; 0; 1i-recursive sequence of degree 2. èbè The sequence 8; 11; 16; 23; 32; 43; 56;::: èi-s-t 70, Form A1, Item 119, see Amthauer, 1973è can be characterized by the rule: "Adding 3 to the ærst member, 5 to the second, 7 to the third, and so on..." However, this sequence is h2; 2;,1i-recursive of degree 2. ècè The sequence 2; 2; 3; 3; 5; 5; 8; 8;::: èkft, Form B, Item 32, see Heller, Gaedike & Weinlíader, 1976è that may be regarded as a slight variant of the type shown in bè is easily recognized as h,1; 0; 0; 0; 3i-recursive of degree 4. Up until now we have been dealing with inænite sequences. However, common number sequence tasks applied in psychological testing and research present only a ænite sequence of some numbers and demand regular continuation of this sequence. We denote such a ænite n-term sequence a 1 ;a 2 ; :::; a n by èa i è n and extend the deænition of inænite linear-recursive sequences in 2.2 to n-term sequences. 2.4 Deænition. Let èa i è n 2 IQ n be a ænite, non-constant n-term sequence a 1 ;a 2 ; :::; a n of rational numbers. Suppose that for some r 2 f1; :::; n, 1g the equations of form è1è for i = r +1;r+2; :::; n hold for some F r := h c; f 1 ; :::; f r i2 D r ç IQ r æèiqnf0gè. Then also èa i è n is said to be h c; f 1 ; :::; f r i-recursive èresp. F r -recursiveè èof length nè. The other notions of 2.2 are used accordingly. The usual number sequence tasks demand that on the basis of a recognized regularity the number sequence is to be "continued" or "completed" by determining one or more subsequent members of the sequence. To make this demand more precise we introduce the notion of an "extension" of a given ænite èlinear-recursiveè sequence. 2.5 Deænition. Let the n-term sequence èa i è n 2 IQ n for some r 2f1; :::; n,1g be F r -recursive with some F r := h c; f 1 ; :::; f r i2 IQ r æèiqnf0gè. If for p = n + k

7 Klaus Korossy: Linear-Recursive Number Sequence Tasks 49 èk; p 2 INè the p-term sequence èa i è p 2 IQ p is also F r -recursive and if the initial n members coincide with èa i è n, then èa i è p is called the k-fold F r -extension of èa i è n. 2.6 Examples. èaè For the 6-term sequence 1; 1; 2; 3; 5; 8 èsee 2.1 ècèè, the sequence 1; 1; 2; 3; 5; 8; 13 ; 21 is a twofold h0; 1; 1i-extension. èbè For the 6-term sequence 1; 3; 4; 4; 3; 1 èsee 2.1 èdèè, the sequence 1; 3; 4; 4; 3; 1;,2 ;,6;,11 is a threefold h,1; 2;,1i-extension. ècè For the 7-term sequence 9; 7; 10; 8; 11; 9; 12 èsee 2.3 èaèè, the sequence 9; 7; 10; 8 ; 11; 9; 12; 10 is a onefold h1; 0; 1i-extension. èdè For the 8-term sequence 2; 2; 3; 3; 5; 5 ; 8; 8 èsee 2.3 ècèè, the sequence 2; 2; 3; 3; 5; 5 ; 8; 8; 14 is a onefold h,1; 0; 0; 0; 3i-extension. Note at this point a serious problem uncovered by 2.6 èdè: The ninth member of the sequence, 14, suggested by the onefold h,1; 0; 0; 0; 3i-extension is not found among the multiple choice answers provided in the test èkft, Form B, Item 32è. Actually, the sequence could also be continued by 12, which is suggested by the sequence rule apparently intended by the test constructors. In this paper the problems of non-unique solutions to number sequence tasks are extensively discussed within the class of linear-recursive number sequences. 3 General type of task, solvability and uniqueness of solution The further considerations in this paper concern number sequence tasks where a ænite 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 deænined notions, we ærst 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 speciæed in the task instruction èlinear recursion formulaeè, 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 D r := IQ r æèiqnf0gè. 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 ènon-constantè n-term sequence èa i è n 2 IQ n of rational numbers a 1 ;a 2 ; :::; a n, a maximal permissible degree r max én of recursion, and the number k for a k-fold extension of èa i è n.

8 Klaus Korossy: Linear-Recursive Number Sequence Tasks 50 I. Find r 2f1; :::; r max g and F r = h c; f 1 ; :::; f r i2d r = IQ r æèiqnf0gè so that èa i è n is F r -recursive. II. Calculate a n+1; :::; a n+k 2 IQ so that èa i è n+k is the k-fold F r -extension of èa i è n. One easily sees that for r max = n, 1 each n-term sequence is linear-recursive of degree r := r max = n, 1 because a solution h c; f 1 ; :::; f r i2 IQ r æèiqnf0gè for is given, for instance, by a n = f 1 a n,1 + ::: + f r a 1 + c f 1 = f 2 = ::: = f r,1 =0; f r 2 IQnf0g arbitrary ; c = a n, f r a 1 : Generally, when some recursion formula for a given sequence is found, the subsequent members can be calculated. Not for each sequence èa i è n 2 IQ n, however, does there exist an r 2 f1; :::; n, 2g such that èa i è n is F r -recursive with F r 2 IQ r æèiqnf0gè èexample: The sequence 1; 1; 1; 1; 1; 2 is not linear-recursive of some degree ré5è. Therefore, the mathematical analysis of a number sequence task basically involves examining the problem of solvability. 3.2 The problem of solvability. Given a ènon-constantè n-term number sequence èa i è n 2 IQ n and a maximal permissible degree r max én 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 r 2f1; :::; r max g. 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 èa i è n 2 IQ n be an n-term sequence for which an F r 2 IQ r æèiqnf0gè exists such that èa i è n is F r -recursive and èa i è n+k 2 IQ n+k is the k-fold F r -extension. I. Are r and F r uniquely determined? If that is the case, then also èa i è n+k is èvia r, F r è uniquely determined. II. Suppose, r; F r are not unique, i.e. for some r;s 2f1; :::; r max g there exist F r ç IQ r æèiqnf0gè and G s ç IQ s æèiqnf0gè, F r 6= G s, such that èa i è n is F r -recursive aswell as G s -recursive, and let èa i è n+k 2 IQ n+k be the F r - extension and èa 0 i èn+k 2 IQ n+k be the G s -extension of èa i è n. The question that needs to be answered is: Are èa i è n+k and èa 0 iè n+k identic? The formulation of the uniqueness problem indicates fundamental problems in the application of number sequence tasks in psychology that are by no means artiæcial:

9 Klaus Korossy: Linear-Recursive Number Sequence Tasks 51 æ That, in fact, for a certain number sequence èa i è n included in a task of type 3.1 there may exist several diæering solution formulae has already been demonstrated by the example sequence in 2.1 èaè: The arithmetic sequence 1; 3; 5; 7; 9; 11; ::: is h2; 1i-recursive aswell as h0; 2;,1i-recursive. In this case, fortunately, both recursion formulae produce identic extensions. Admittedly, at least for those versions of sequence tasks where only the completion of the sequence is asked for, one might argue that whenever the continuation of a number sequence is unique or "face-valid", the uniqueness problem would be of no concern. This evaluation, however, could prove to be rather careless because the use of diæerent ècomplexè formulae may lead to diæerent solution times und thus to inestimable biases in test scores. Moreover, particularly in empirical investigations testing cognitive-processes models one can hardly disregard which one of several possible formulae is used in a single case. æ Even more problematic for psychometric testing as well as for psychological experimentation are the cases addressed by 3.3 II. where diæerent formulae lead to diæerent extensions of a given sequence. Indeed, this case may occur in number sequence tasks of type 3.1. As an example consider the sequence 2; 2; 2; 4; 6; 10 ; this sequence is, for instance, h,2; 1; 1; 1i-recursive with the term a 7 = 18, h2; 1; 1;,1i-recursive with the term a 7 = 14, h,1; 1; 1; 1 2 i-recursive with the term a 7 = 17, h1; 1; 1;, 1 2 i-recursive with the term a 7 =15. 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 D r 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. 4 Existence and uniqueness of solution formulae In this section, we will derive some basic results concerning the problems of existence and uniqueness of recursion formulae for given number sequences. For this purpose we refer to the theory of linear equation systems, that provides several useful theorems on the existence, uniqueness and structure of solutions of a given system of linear equations. Let us ærst introduce some notation. 4.1 Notation. Recall that, according to 2.2è2.4, a ènon-constantè number sequence èa i è n 2 IQ n is F r -recursive with some F r := h c; f 1 ; :::; f r i 2 D r ç IQ r æèiqnf0gè èr 2f1; :::; n, 1gè if and only if the linear equation system èlesè a i = rx j=1 f j a i,j + c; i = r +1;r+2; :::; n è1è

10 Klaus Korossy: Linear-Recursive Number Sequence Tasks 52 of n, r equations in r +1 variables is satisæed. Equivalently è1è can be represented by the matrix equation a n,1 a n,2 æææ a n,r 1 a n,2 a n,3 æææ a n,r, a r+1 a r æææ a 2 1 a r a r,1 æææ a f 1 f 2. f r c = a n a n,1. a r+2 a r : è2è If we denote the èn, rè æ r-matrix on the left-hand side of è2è èwhich is usually called the coeæcient matrix of è2èè by A and the vector on the right-hand side of è2è by ~a, then the LES è1è can brieæy be represented by the augmented matrix èaj~aè := a n,1 a n,2 æææ a n,r 1 a n a n,2 a n,3 æææ a n,r,1 1 a n, a r+1 a r æææ a 2 1 a r+2 5 : è3è a r a r,1 æææ a 1 1 a r Example. Given the 5-term sequence 1; 1; 1; 3; 5. Is this sequence linearrecursive of some degree r =2orr =3? The approach for r = 2 and F 2 = h c; f 1 ;f 2 i leads to the LES 5 = 3f 1 +1f 2 + c 3 = 1f 1 +1f 2 + c 1 = 1f 1 +1f 2 + c with the augmented-matrix representation èaj~aè := The approach for r = 3 and F 3 = h c; f 1 ;f 2 ;f 3 i yields the LES 5 = 3f 1 +1f 2 +1f 3 + c 3 = 1f 1 +1f 2 +1f 3 + c with the augmented-matrix representation ç èaj~aè := We leave the solution to our example task for later analysis. 3 5 : è4è ç : è5è Now let be given a number sequence task of the general type 3.1 including some non-constant number sequence èa i è n 2 IQ n and some æxed recursion degree r 2 f1; :::; r max g. Our purpose is to derive some general statements about existence and uniqueness of recursion formulae for èa i è n within the domain D r := IQ r æèiqnf0gè. From the theory of LESs several useful results are available about existence and type of solution to è1è based on the ranks 3 of the coeæcient matrix and the 3 Recall that the rank of a matrix A is deæned as the maximal number of linearly independent row-vectors or column-vectors of A, respectively. Denotation: rank èaè.

11 Klaus Korossy: Linear-Recursive Number Sequence Tasks 53 augmented matrix in è3è of the LES è1è. We summarize some useful statements from the theory of LESs in the following lemma èfor the theory of LES see e.g. Kowalsky, 1974; Kuiper, 1962è. However observe, that the statements refer to solutions of è1è within IQ r+1 ; thus, an important step will be to modify the statements such that they apply to the domain D r = IQ r æèiqnf0gè for recursion formulae of degree r. 4.3 Lemma. Consider for a given number sequence èa i è n the LES è1è and the corresponding augmented-matrix representation è3è. For solvability and solutions of è1è in IQ r+1 the following statements hold: èaè The LES è1è is solvable with some F r 2 IQ r+1 if and only if rank èaè = rank èaj~aè. èbè Assume the system è1è is solvable in IQ r+1. determined if and only if rank èaè =r +1. Then the solution is uniquely ècè The system è1è has inænitely many solutions in IQ r+1 if and only if rank èaj~aè = rank èaè ér+1. Let d := r +1, rank èaè ; then in the solution formulae d parameters can be chosen arbitrarily. We transfer the statements of 4.3 to the problems of existence and uniqueness of solution formulae for è1è within the domain D r = IQ r æèiqnf0gè.the essential point hereby is the condition that by Deænition 2.2 the recursion coeæcient f r in each considered solution formula must not be equal to Theorem. Let be given a non-constant n-term number sequence èa i è n 2 IQ n of rational numbers and some æxed recursion degree r 2 f1; :::; r max g. For the existence and uniqueness of recursion formulae in the domain IQ r æèiqnf0gè satisfying the LES è1è, the following statements hold: èaè If in the augmented-matrix representation è3è of the LES è1è holds rank èaè é rank èaj~aè, then the sequence èa i è n is not F r -recursive for any F r 2 IQ r æèiqnf0gè èi.e. not linear-recursive of degree rè. èbè Assume the sequence èa i è n is F r -recursive for some F r 2 IQ r æèiqnf0gè. Then the formula F r is unique for èa i è n if and only if in è3è holds rank èaè = r +1. ècè Assume the sequence èa i è n is F r -recursive for some F r 2 IQ r æèiqnf0gè. Then there exist even inænitely many recursion formulae for èa i è n in IQ r æèiqnf0gè if and only if rank èaè ç rank èaj~aè é r +1. Let d := r +1, rank èaè; then in the recursion formulae d elements can be chosen arbitrarily èwith the restriction that if f r is an arbitrary parameter, then it only can be chosen from IQnf0g è. Proof. èaè The LES è1è is unsolvable in IQ r æèiqnf0gè if and only if it is either unsolvable in IQ r+1 or there only existèsè èone or inænitely manyè solutionèsè of è1è in IQ r æf0g ; therefore from the equivalence 4.3 èaè remains only a necessary condition for the existence of a recursion formula F r for a given sequence èa i è n within IQ r æèiqnf0gè. The contraposition yields èaè. èbè The LES è1è is uniquely solvable with some F r := h c; f 1 ; :::; f r i2 IQ r æèiqnf0gè if and only if F r is a unique solution of è1è in IQ r+1 with f r 6= 0; but if there exists

12 Klaus Korossy: Linear-Recursive Number Sequence Tasks 54 a solution formula F r 2 IQ r æèiqnf0gè for è1è, then 4.3 èbè includes a necessary and suæcient condition for the uniqueness of F r. ècè The LES è1è is non-uniquely solvable with formulae of type F r := h c; f 1 ; :::; f r i 2 IQ r æèiqnf0gè if and only if there exist inænitely many solutions of type F r to è1è in IQ r+1 where either f r 6= 0 is æxed or arbitrarily choosable in IQnf0g. Therefore, if a solution formula F r 2 IQ r æèiqnf0gè for è1è exists at all, then 4.3 ècè reveals conditions and structure of an inænitely large solution set Examples. èaè Given the 5-term sequence 1; 1; 1; 3; 5 from 4.2. According to 4.4 èaè, this sequence is not F 2 -recursive for an F 2 = h c; f 1 ;f 2 i 2 D 2, because for the augmented matrix è4è holds rank èaè = 2 é 3 = rank èaj~aè. èbè Given the 5-term sequence 1; 1; 2; 3; 5. This sequence is easily recognized as h0; 1; 1i-recursive èwith degree r = 2è. According to 4.4 èbè, the recursion formula is unique because in the corresponding augmented-matrix representation holds rank èaè ç rank èaj~aè = = 3. ècè The 5-term sequence 1; 1; 1; 3; 5 from 4.2. is for degree r = 3 for instance h0; 1; 0; 2i-recursive. The general solution approach with r = 3 shows that rank èaè ç rank èaj~aè = 2 é 4. Actually one obtains recursion formulae of the type h 2, f 2, f 3 ;1;f 2 ; f 3 i where f 2 2 IQ and f 3 2 IQ nf0g are arbitrary parameters. èdè For the 5-term sequence 1; 3; 3; 3; 3 the general solution approach with r =3 shows that rank èaè ç rank èaj~aè = 2 é 4. However, one obtains only solutions of the type h 3, 3f 1, 3f 2 ; f 1 ; f 2 ;0i where f 2 and f 3 are arbitrary parameters, but f 3 = 0 is æxed. Thus the given sequence is according to Deænition 2.2 not linear-recursive ofdegree 3. This example proves that in 4.4 ècè the existence of some permissible recursion formula is a necessary precondition. From 4.4 èbè we derive an immediate consequence. If in è3è holds rank èaè = r +1, then we can conclude from r +1 = rank èaè ç minfn, r;r +1g that n, r ç r +1, and arrive at a useful necessary condition for the uniqueness of a recursion formula. 4.6 Corollary. Given an n-term number sequence èa i è n. A necessary condition for the uniqueness of a recursion formula F r 2 IQ r æèiqnf0gè for èa i è n is the condition r ç 1 2 èn, 1è. The statement of 4.6 expressed in another way: If for a recursion degree r é 1 2èn, 1è there exist recursion formulae at all, then in fact inænitely many, the structure of which is described in 4.4 ècè. Observe, however, that the condition given in 4.6 is not suæcient asisveriæed by the following example. 4.7 Example. The 6-term sequence 1; 2; 3; 4; 5; 6 is linear-recursive of degree r = 2 with inænitely many recursion formulae of type h 1+f 2 ;1, f 2 ; f 2 i with f 2 2 IQnf0g arbitrary, although the necessary condition in 4.6 is fulælled.

13 Klaus Korossy: Linear-Recursive Number Sequence Tasks Remark. Transform for r ç n, 2 the augmented matrix è3è of the LES è1è equivalently by subtracting each row from the preceding one èand leaving the last row unchangedè, and set as an abbreviation d i := a i, a i,1 è i =2; 3; :::; n è for the diæerence of each two succeeding members of the sequence; then we obtain the matrix d n,1 d n,2 æææ d n,r 0 d n d n,2 d n,3 æææ d n,r,1 0 d n, d r+1 d r æææ d 2 0 d r+2 a r a r,1 æææ a 1 1 a r+1 3 ç 7 5 =: D ~o d ~ a r ::: a 1 1 a r+1 ç : è6è Now, Theorem 4.4 can be applied to èdj ~ dè. For instance, one obtains the statement: If the sequence èa i è n is F r -recursive for some F r 2 IQ r æèiqnf0gè, then the formula F r is uniquely determined if and only if in è6è holds rank èdè =r. This consideration is interesting insofar as it shows that the problems of existence and uniqueness of a recursion formula for a given number sequence can be decided solely by referring to the sequence of diæerences and hence do not depend on the recursion constant. We will refer back to this in Section 5. Up to now we have derived several statements on the existence, uniqueness, and structure of a solution set for è1è in IQ r æèiqnf0gè under the precondition of a æxed degree r 2 f1; :::; r max g of recursion. Suppose now we had a certain result for a æxed recursion degree r. What can we conclude for solutions of recursion degrees sérand sér? For sérthe following theorem is provable. 4.9 Theorem. Assume the n-term sequence èa i è n is F r -recursive for some F r 2 IQ r æèiqnf0gè with rén,1. Then for each recursion degree s with résén there exist inænitely many other recursion formulae F s 2 IQ s æèiqnf0gè such that èa i è n is F s -recursive. Proof. Let èa i è n be F r -recursive with F r := h c; f 1 ; :::; f r i2 IQ r æèiqnf0gè, that is, the following equations are satisæed: a i = f 1 a i,1 + :::+ f r a i,r + c; i = r +1;r+2;:::;n: Let r = 1. For each z 2 IQnf0g and i = r + 2; :::; n we can transform in the following way: a i = f 1 a i,1 + c = èf 1, zèa i,1 + za i,1 + c = èf 1, zèa i,1 + zèf 1 a i,2 + cè+c = èf 1, zèa i,1 + zf 1 a i,2 +èz +1èc =: f1 æ a i,1 + f2 æ a i,2 + c æ Let r ç 2. For each z 2 IQnf0g and i = r +2; :::; n we transform the following way: a i = f 1 a i,1 + f 2 a i,2 + :::+ f r a i,r + c = èf 1, zèa i,1 + za i,1 + f 2 a i,2 + :::+ f r a i,r + c = èf 1, zèa i,1 + zèf 1 a i,2 + :::f r a i,r,1 + cè+f 2 a i,2 + :::+ f r a i,r + c = èf 1, zèa i,1 +èzf 1 + f 2 èa i,2 + :::+èzf r,1 + f r èa i,r + zf r a i,r,1 +èz +1èc =: f æ 1 a i,1 + f æ 2 a i,2 + :::+ f æ r a i,r + f æ r+1 a i,r,1 + c æ

14 Klaus Korossy: Linear-Recursive Number Sequence Tasks 56 Because z 2 IQnf0g can be arbitrarily chosen, there exist inænitely many recursion formulae h c æ ;f1 æ ; :::; f r+1 æ i in IQr+1 æèiqnf0gè. This argument can be iterated for higher recursion degrees until s = n, 1 inclusively Example. Given the èuniquelyè h3;,1;,1i-recursive 6-term sequence 1; 2; 0; 1; 2; 0. Using the procedure from the proof of Theorem 4.9, this sequence turns out to be also h 3èz + 1è;,1, z;,z, 1;,z i-recursive with arbitrary z 2 IQnf0g. Fixing, for instance, z :=,1 one obtains the recursion formula h0; 0; 0; 1i where the constant is 0. The example in 4.10 may demonstrate that recursion formulae of a higher degree need not necessarily be more complicated in a psychological sense than the recursion formulae of a lower degree. For instance, æxing the introduced parameter z :=,1 in the proof to Theorem 4.9 èsee 4.10è leads to the constant c? = 0; thus, the inhomogeneous diæerence equation a i = f 1 a i,1 + :::+ f r a i,r + c; i = r +1; :::; n transforms into a homogeneous diæerence equation a i = f? 1 a i,1 + :::+ f? r a i,r,1 ; i = r +2;:::;n: We document this consideration in the following corollary Corollary. Each h c; f 1 ; :::; f r i-recursive sequence èa i è n of degree rén,1 with c 6= 0 is also h 0;f æ 1 ; :::; f æ r+1 i-recursive of degree r + 1 with constant c? =0. If one knows the solution structure for the LES è1è in the domain IQ r æèiqnf0gè for some æxed r, then there are also conclusions for the existence and uniqueness of solution formulae of degree s é r available. From Theorem 4.9 we obtain by contraposition the following theorem Theorem. If for some n-term sequence èa i è n there exists either none or only exactly one recursion formula of some degree r, then there exist no recursion formulae for èa i è n of some degree sér. Note that the condition in Theorem 4.12 is suæcient but not necessary. For a sequence èa i è n there may exist inænitely many recursion formulae of some degree r, but no recursion formula of degree s = r, 1. This is exempliæed by the 5- term sequence 1; 1; 1; 3; 5 from example 4.2. In 4.5 èaè this sequence has been proven not to be linear-recursive of degree 2; in 4.5 ècè, however, it turned out that for this sequence there exist inænitely many recursion formulae of degree 3. Nevertheless, under certain speciæc conditions the conclusion from inænitely many recursion formulae of some degree r to recursion formulae of lower degree is valid. This is stated by the easily comprehensible theorem in Theorem. Assume for the sequence èa i è n and some æxed recursion degree r there exist inænitely many recursion formulae of the type F r := h c; f 1 ; :::; f r i2 IQ r æèiqnf0gè, and let d := r +1, rank èaè ç 1 be the number of arbitrary parameters in F r. Then there exists a recursion formula F s 2 IQ s æèiqnf0gè for sérsuch that èa i è n is F s -recursive, when the following two conditions are satisæed:

15 Klaus Korossy: Linear-Recursive Number Sequence Tasks 57 èiè Among the formulae of type F r there exists some formula with f s 6=0 and f s+1 = ::: = f r =0. èiiè Additionally to the recursion equations for i = r +1; :::; n, which are satisæed by the formulae of type F r, the linear equations a i = rx j=1 f j a i,j + c; for i = s +1; :::; r are fulælled by the formula F s = h c; f 1 ; :::; f s i2 IQ s æèiqnf0gè Examples. èaè The 5-term sequence 1; 2; 2; 2; 4 is for r = 3 linear-recursive with inænitely many recursion formulae of type F 3 := h,2f 1,2f 2 ; f 1 ; f 2 ;2i with f 1 ;f 2 2 IQ arbitrary. But f 3 = 2 is æxed and cannot be set equal to 0. Thus condition èiè of Theorem 4.13 is not fulælled. In fact, the sequence is not linear-recursive of degree r =2. èbè In 4.7 the 6-term sequence 1; 2; 3; 4; 5; 6 was shown to be linear-recursive of degree r = 2 with inænitely many recursion formulae of type F 2 := h 1+ f 2 ;1, f 2 ; f 2 i with f 2 2 IQnf0g arbitrary. In F 2 set the parameter f 2 equal to 0 and obtain the formula F 1 := h1; 1i. The recursion equation for a 2, 2=1æ 1 + 1, is fulælled. Hence the sequence is also F 1 -recursive Consequences. The results concerning the problems of solvability and ènon-èuniqueness of solutions to the general type of number sequence task in 3.1 do not depend on the domain for the members of a number sequence. Instead, the investigated solvability and uniqueness problems are caused decisively by the fact that neither for the recursion degree r nor for the domain D r of permitted recursion formulae essential restrictions have been introduced. This is in accordance with the usual formulation of number sequence tasks in psychological applications. In order to avoid uncontrolled non-uniqueness, three ènonexclusiveè approaches seem suggested: aè The restriction of the maximal permissible recursion degree r max, depending on the length n of the number sequence, by imposing the condition r max ç èn, 1è. 1 2 bè The restriction of the domain D r of permissible recursion formulae for each permitted r. cè The very careful construction of each number sequence task and analysis of the solution formulae for the sequence by means of a LES è1è. 5 A special type of linear-recursive number sequence tasks Suppose that for a psychological investigation, tasks including linear-recursive 6- term number sequences of maximal recursion degree r = 3 are to be utilized; the

16 Klaus Korossy: Linear-Recursive Number Sequence Tasks 58 sequences shall not be constant, and their members as well as the continuations sought shall consist of natural numbers. The problem to analyze is how to construct appropriate tasks that are solvable at all and, moreover, that are solvable by some uniquely determined solution. The problem introduced above actually occurred in a certain research context of knowledge psychology 4. We will analyze this speciæc problem with the purpose of carrying out an exemplary mathematical task-type analysis with respect to the basic problems of solvability and uniqueness of solution. Apparently for a ènon-constantè 6-term sequence èa i è 6 the necessary condition of Corollary 4.6 for the uniqueness of a recursion formula within D 3 = IQ 3 æèiqnf0gè is not satisæed because r =3é2:5 = 1 2è6, 1è. From that follows: If there exist recursion formulae within D 3 then there even exist inænitely many; furthermore, in this case, there may exist also èpossibly inænitely manyè recursion formulae in D 2 = IQ 2 æèiqnf0gè, and possibly even in D 1 = IQæè IQnf0gè. Moreover, according to Theorem 4.9 each number sequence that is linear-recursive ofdegree 1 or 2 is also linear-recursive with inænitely many recursion formulae of the higher degreeèsè until 3 inclusively. Restricting the maximal permissible recursion degree for 6-term number sequences to r = 3 does not provide suæcient control of the ènon-èuniqueness problem. Therefore, according to the approach proposed in 4.15 bè we additionally impose some restrictions for the domain D r of permissible recursion formulae. From a psychological viewpoint the following restriction for D r is suggested: 5.1 Restriction èrè. èiè The recursion formulae have the maximal degree r max =3. èiiè In each recursion formulae, whatever the degree 1 ç r ç 3may be, at most one recursion coeæcient is unequal to 1. èapart from that, the constant can take arbitrary rational values.è Formally, the Restriction èrè implies that only the following domains for the recursion formulae of diæerent degrees r 2f1; 2; 3g are to be taken into account: Degree r =1: D 1 := IQ æ è IQnf0g è Degree r =2: D 1 2 := ff 2 j F 2 = hc; f 1 ; 1i 2 IQ 3 ;f 1 6=1g D 2 2 := ff 2 j F 2 = hc; 1;f 2 i2 IQ 3 ;f 2 6=0g Degree r =3: D 1 3 := ff 3 j F 3 = hc; f 1 ; 1; 1i 2 IQ 4 ;f 1 6=1g D 2 3 := ff 3 j F 3 = hc; 1;f 2 ; 1i 2 IQ 4 ;f 2 6=1g D 3 3 := ff 3 j F 3 = hc; 1; 1;f 3 i2 IQ 4 ;f 3 6=0g The restriction f 1 6=0 in D 1 2 is introduced only for the purpose that a formula hc; 1; 1i can be assigned uniquely to the domain D 2 2 with the consequence that D 1 2 ë D2 2 = ; ; the restrictions for D 1 3 and D 2 3 are introduced correspondingly. Thus, the above domains are pairwise disjoint. Furthermore, the following abbreviations will be used in this section: D 2 := D 1 2 ë D2 2 4 We refer to the study mentioned in footnote 2.

17 Klaus Korossy: Linear-Recursive Number Sequence Tasks 59 D 3 := D 1 3 ë D2 3 ë D3 3 D := D 1 ë D 2 ë D 3 Observing these conventions, we next formulate the special type of number sequence task obtained from the general type of task in 3.1 by imposing the restrictions for the recursion degree and the permitted recursion formulae. 5.2 Special type of number sequence task. Given some ènon-constantè 6-term sequence èa i è 6 2 IN 6 of natural numbers a 1 ;a 2 ; :::; a 6 and the maximal permissible degree r max =3 of recursion. I. Find r 2f1; 2; 3g and F r = h c; f 1 ; :::; f r i2d r so that èa i è 6 is F r -recursive. II. Compute a 7 2 IN so that èa i è 7 is the onefold F r -extension of èa i è 6. As remarked following 3.2, the solvability for some number sequence task is usually ensured by constructing the sequence according to a certain permitted formula. Thus the more serious problem in our context is the ènon-èuniqueness problem for the solutions. The following analyses therefore concentrate on the ènon-èuniqueness problem for our special type of number sequence task. 5.3 The problem of uniqueness of solution. Let èa i è 6 2 IN 6 be a 6-term sequence. Assume that for some r 2f1; 2; 3g there exists an F r 2 D r such that èa i è 6 is F r -recursive and èa i è 7 2 IN 7 is the onefold F r -extension of èa i è 6. I. Are r and F r uniquely determined? If that is the case, then also èa i è 7 is èvia r, F r è uniquely determined. II. Suppose, r; F r are not unique, i.e. for some r;s 2 f1; 2; 3g there exist F r ç D r and G s ç D s with F r 6= G s, such that èa i è 6 is F r -recursive as well as G s -recursive, and let èa i è 7 2 IN 7 be the F r -extension and èa 0 i è7 2 IN 7 be the G s -extension of èa i è 6. The question that must be answered is: Are èa i è 7 and èa 0 i è7 identic? Introducing the apparently rather strong Restriction èrè for the domain of permissible recursion formulae may suggest two hypotheses concerning our ènonèuniqueness problem for solutions í a "strong" hypothesis and a "weak" hypothesis. 5.4 Hypotheses. I. Because of the Restriction èrè, for each sequence èa i è 6 2 IN 6 holds: If èa i è 6 is F -recursive for some F 2 D at all, then F is uniquely determined èand thus the F -extension of èa i è 6 as wellè. If this hypothesis proves to be false, then at least:

18 Klaus Korossy: Linear-Recursive Number Sequence Tasks 60 II. If for a given sequence èa i è 6 2 IN 6 there exist F; F 0 2 D with F 6= F 0 such that èa i è 6 is F -recursive aswell as F 0 -recursive, and if èa i è 7 2 IN 7 is the F -extension and èa 0 i è7 2 IN 7 the F 0 -extension of èa i è 6, then a 7 = a 0 7 holds. With these hypotheses as guidelines we will analyze the uniqueness problem in the following section. 6 Analysis of the uniqueness problem for the special type of number sequence task In this section, we will show that both hypotheses formulated at the end of Section 5 are false. The following proposition disproves Hypothesis 5.4 I. 6.1 Proposition. Each hc; f 1 i-recursive number sequence èa i è 6 2 IN 6 with hc; f 1 i2d 1 = IQ æ è IQnf0g è is additionally èaè hc æ ; f æ 1 ;1i-recursive for appropriate hc æ ; f æ 1 ;1i2D 1 2 ; èbè hc? ; f? 1 ;1;1i-recursive for appropriate hc? ; f? 1 ;1;1i2D 1 3 ; ècè hc æ ;1;f æ 2 i-recursive for appropriate hcæ ;1;f æ 2 i2d2 2 in the case that f 1 6=1; èdè hc? ;1;f? 2 ;1i-recursive for appropriate hc? ;1;f? 2 ;1i2D 2 3 ; èeè hc æ ;1;1;f æ 3 i-recursive for appropriate hcæ ;1;1;f æ 3 i2d3 3. Proof. Let èa i è 6 2 IN 6 be some hc; f 1 i-recursive sequence with hc; f 1 i2d 1. èaè For i =3; 4; 5; 6 holds: a i = f 1 a i,1 + c = èf 1, 1 f 1 èa i,1 + 1 f 1 èf 1 a i,2 + cè+c = èf 1, 1 f 1 èa i,1 + a i,2 +è 1 f 1 +1èc =: f æ 1 a i,1 + a i,2 + c æ with f æ 1 6=1 for f 1 2 IQnf0g. èbè For i =4; 5; 6 holds: a i = f 1 a i,1 + c ç 1+ 1 f1 = f 1, f 1 = = ç 1+ 1 f1 f 1, f 1 ç 1+ 1 f1 f 1, f 1 ç a i, f1 èf 1 a i,2 + cè+c f 1 ç a i,1 + ç 1+ 1 f1 = f 1, f 1 =: f 1? a i,1 + a i,2 + a i,3 + c? ç1+ 1 f1 ç a i,2 + ç 1+ 1 f1 f 1 +1 c ç a i,1 + a i,2 + 1 èf 1 a i,3 + cè+ç f1 +1+f1 2 f 1 f1 2 ç ç a i,1 + a i,2 + a i, ç 2 c f 1 with f? 1 6=1 for f 1 2 IQnf0g. ènote that the case f 1 =,1 need not be excluded!è ç ç c