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Linear-recursive number sequences: definitions and examples

Many number sequences have the characteristic property that subsequent members are related to the preceding members by linear equations.

2.1 Examples.

(a)
Arithmetic sequences as, for instance, the sequence tex2html_wrap_inline1451 can be described by giving the first member tex2html_wrap_inline1453 (here: tex2html_wrap_inline1455) and the general term tex2html_wrap_inline1457 for tex2html_wrap_inline1459 (here: d = 2); another way of description is by giving tex2html_wrap_inline1453 and the recursive equation
displaymath1441
or, for instance, the initial members tex2html_wrap_inline1465 and the recursive equation
displaymath1442
(b)
Geometric sequences as, for instance, the sequence tex2html_wrap_inline1467 can be described by giving the first member tex2html_wrap_inline1453 (here: tex2html_wrap_inline1471) and the general term tex2html_wrap_inline1473 for tex2html_wrap_inline1459 (here: tex2html_wrap_inline1477), but also recursively by giving tex2html_wrap_inline1453 and the equation
displaymath1443
(c)
A well-known sequence of numbers is the Fibonacci sequence tex2html_wrap_inline1481 with the recursive description
displaymath1444
(d)
The sequence tex2html_wrap_inline1483 is recursively described by
displaymath1445

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 first introduce some basic concepts concerning the special type of number sequences shown in the examples. The following abbreviations will be used:

tex2html_wrap_inline1485 the set of natural numbers tex2html_wrap_inline1487 (without 0)
tex2html_wrap_inline1489 the set of rational numbers
tex2html_wrap_inline1491 the set of n-tupel of elements of set A

2.2 Definition. Let tex2html_wrap_inline1497 be a (infinite) non-constant sequence tex2html_wrap_inline1499 of rational numbers. If for a given tex2html_wrap_inline1501 there exists an (r+1)-tupel tex2html_wrap_inline1505 with tex2html_wrap_inline1507 such that the recursive equations
equation145
are satisfied, then the sequence tex2html_wrap_inline1497 is called linear-recursive of degree r with the recursion coefficients tex2html_wrap_inline1513 and the constant c, in brief tex2html_wrap_inline1517-recursive (or tex2html_wrap_inline1519-recursive). The (r+1)-tupel tex2html_wrap_inline1517 is called a recursion formula (of degree r) for tex2html_wrap_inline1497, occasionally a solution to (1); tex2html_wrap_inline1527 (which may be a proper subset of tex2html_wrap_inline1529 ) is called the domain for the recursion formulae of degree r.

Remarks.

(i) Within the calculus of differences, (1) describes an inhomogeneous linear difference equation with constant coefficients and constant inhomogenity (see e.g. Markuschewitsch, 1955; Rommelfanger, 1986, pp. 36-37), where, however, it is permitted that the recursion equation (1) may be satisfied beginning only with the members of a higher index than r+1. Because our consideration differs from the typical way of looking at the problems in the calculus of difference, we will not pursue that approach here.

(ii) 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 coefficients, and the constant c are elements of the field IQ of rational numbers which is closed under the elementary arithmetic operations. Special attention will later be given to the domain tex2html_wrap_inline1539 for the recursion formula.

(iii) In Definition 2.2 the condition tex2html_wrap_inline1507 is to prevent that for instance the sequence tex2html_wrap_inline1543, with tex2html_wrap_inline1545 for tex2html_wrap_inline1547, may be classified as tex2html_wrap_inline1549-recursive. According to 2.2 this sequence is not tex2html_wrap_inline1549-recursive, since the condition tex2html_wrap_inline1553 is not satisfied. However, this sequence is also not tex2html_wrap_inline1555-recursive of degree 1, because the recursion equation does not hold for tex2html_wrap_inline1557 (see Remark (i)). Instead the sequence is, for instance, tex2html_wrap_inline1559-recursive of degree 4, because tex2html_wrap_inline1561 holds for tex2html_wrap_inline1547.

(iv) 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.

Examples. In the examples of 2.1, the sequence (a) is tex2html_wrap_inline1565-recursive as well as tex2html_wrap_inline1567-recursive; the sequence (b) is tex2html_wrap_inline1569-recursive; the Fibonacci sequence (c) is tex2html_wrap_inline1571-recursive; the sequence (d) is tex2html_wrap_inline1573-recursive.

As is seen in the examples above certain sequences can be represented by several different recursion formulae. On the other hand of course, each recursion formula describes infinitely many sequences depending on the given initial members.

Perhaps the concept of a linear-recursive number sequence seems to be relatively specific. 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 first glance. The following examples may serve to verify this.

2.3 Examples.

(a)
The sequence tex2html_wrap_inline1575 (I-S-T 70, Form A1, Aufgabengruppe 06, see Amthauer, 1973) 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 Definition 2.2 as a tex2html_wrap_inline1577-recursive sequence of degree 2.
(b)
The sequence tex2html_wrap_inline1579 (I-S-T 70, Form A1, Item 119, see Amthauer, 1973) can be characterized by the rule: ''Adding 3 to the first member, 5 to the second, 7 to the third, and so on ...'' However, this sequence is tex2html_wrap_inline1581-recursive of degree 2.
(c)
The sequence tex2html_wrap_inline1583 (KFT, Form B, Item 32, see Heller, Gaedike & Weinläder, 1976) that may be regarded as a slight variant of the type shown in b) is easily recognized as tex2html_wrap_inline1585-recursive of degree 4.

Up until now we have been dealing with infinite sequences. However, common number sequence tasks applied in psychological testing and research present only a finite sequence of some numbers and demand regular continuation of this sequence. We denote such a finite n-term sequence tex2html_wrap_inline1589 by tex2html_wrap_inline1591 and extend the definition of infinite linear-recursive sequences in 2.2 to n-term sequences.

2.4 Definition. Let tex2html_wrap_inline1595 be a finite, non-constant n-term sequence tex2html_wrap_inline1589 of rational numbers. Suppose that for some tex2html_wrap_inline1601 the equations of form (1) for tex2html_wrap_inline1603 hold for some tex2html_wrap_inline1605. Then also tex2html_wrap_inline1591 is said to be tex2html_wrap_inline1517-recursive (resp. tex2html_wrap_inline1519-recursive) (of length n). 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 finite (linear-recursive) sequence.

2.5 Definition. Let the n-term sequence tex2html_wrap_inline1595 for some tex2html_wrap_inline1601 be tex2html_wrap_inline1519-recursive with some tex2html_wrap_inline1623. If for tex2html_wrap_inline1625 tex2html_wrap_inline1627 the p-term sequence tex2html_wrap_inline1631 is also tex2html_wrap_inline1519-recursive and if the initial n members coincide with tex2html_wrap_inline1591, then tex2html_wrap_inline1639 is called the k-fold tex2html_wrap_inline1519-extension of tex2html_wrap_inline1591.

2.6 Examples.

(a)
For the 6-term sequence tex2html_wrap_inline1647 (see 2.1 (c)), the sequence tex2html_wrap_inline1649 tex2html_wrap_inline1651 is a twofold tex2html_wrap_inline1571-extension.
(b)
For the 6-term sequence tex2html_wrap_inline1655 (see 2.1 (d)), the sequence tex2html_wrap_inline1657 tex2html_wrap_inline1659 is a threefold tex2html_wrap_inline1573-extension.
(c)
For the 7-term sequence tex2html_wrap_inline1663 (see 2.3 (a)), the sequence tex2html_wrap_inline1665 tex2html_wrap_inline1667 is a onefold tex2html_wrap_inline1577-extension.
(d)
For the 8-term sequence tex2html_wrap_inline1671 tex2html_wrap_inline1673 (see 2.3 (c)), the sequence tex2html_wrap_inline1671 tex2html_wrap_inline1677 is a onefold tex2html_wrap_inline1585-extension.

Note at this point a serious problem uncovered by 2.6 (d): The ninth member of the sequence, 14, suggested by the onefold tex2html_wrap_inline1585-extension is not found among the multiple choice answers provided in the test (KFT, Form B, Item 32). 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.


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Next: General type of task Up: MPR-online 1998Vol.3, No.1 Previous: Number sequence tasks in

Methods of Psychological Research 1998 Vol.3 No.1
© 1998 Pabst Science Publishers