SEQUENCES FOR STUDENT INVESTIGATION

ABSTRACT:

We describe sum of two units classes of sequences that give rise to accessible question at issues for undergraduate research. These point to be solved [i]or[/i] settleds may be understood with virtually no prerequisites and are well suited for computer-aided investigation. The first succession is a variation of individual introduced by Stephen Wolfram in connection with his research of cellular automata. The next to the first is an exploration of a modified "look and say" succession that was introduced by Richard shore and John Conway. For each following we discuss the questions that motivated their application of mind and outline the progress that has been made upon them by undergraduates. Finally, we intimate several open questions for subsequent time student investigation.

KEYWORDS: Cellular automata, direct the eye and say, undergraduate research, succession Ducci.

INTRODUCTION

The view of this article is to address a ne felt by means of many who believe in the value of undergraduate research-the ne for accessible on the contrary sufficiently rich problems for learners to investigate. We suggest a list of like problems below in connection with the application of mind of two classes of successions The sequences are very different, on the contrary share many important features that make them suitable for undergraduate research. They are easy to define and require virtually no prerequisites to understand, and the two sequences have been investigated luckily (though far from exhaustively) by means of undergraduate mathematics majors. A further commonality is that the pair sequences allow students to count simple examples by hand, on the other hand students quickly realize that they can profit by the agency of enlisting the aid of a computer In the case of the first succession a student independently programmed a Microsoft outdo spreadsheet to carry out computations, and in the case of the next to the first sequence, students designed and wrote their have a title to computer program in C.

Of course, the real goal is for scholars to do more than just exhibit a list of examples. scholars should be able to discover abstract general arises consider their data, form theorys based on that data, and finally ascertain some theorems. The students who have worked with the successions presented here have done just that, and examples of their proceeds are included. The final productions of the student research included a 15-page report for a class and a PowerPoint presentation given to the campus community. scholar evaluations of the research throws were favorable.

VARIATIONS ON A following DUE TO WOLFRAM

We begin with an N-digit number, ?±^sub 0^ whose digits consist single of zeros and ones, and we deliver over to such a number as a binary number. We use ?±^sub 0^ to generate a fresh N-digit binary number, ?±^sub 1^ by dint of applying the following algorithm. The ith digit (reading from left to right) of ?±^sub 1^ is assigned the value O if the ith and (i + 1)st digits of ?±^sub 0^ are the same and 1 if the ith and (i + 1)st digits of ?±^sub 0^ are different. To find the Nth digit of ?±^sub 1^ we wrap around and compare the Nth and first digits of ?±^sub 0^ in the same manner. We illustrate the proces with the following example.

Example 1:

Let N=4 and consider ?±^sub 0^=1011 To generate ?±^sub 1^ we proce as follows:

1 To find the first digit of ?±^sub 1^ we examine the first and next to the first digits of ?±^sub 0^. The first digit is 1 and the next to the first is O. Since these are different, we assign the value 1 to the first digit of ?±^sub 1^ : ?±^sub 1^ = 1xxx

2 To find the next to the first digit of ?±^sub 1^ we examine the next to the first and third digits of ?±^sub 0^ The next to the first digit is 0 and the third is 1 Since these are different, we assign the value 1 to the next to the first digit of ?±^sub 1^ : ?±^sub 1^ = 11xx

3 To find the third digit of ?±^sub 1^ we examine the third and fourth digits of ?±^sub 0^ The third digit is 1 and the fourth is 1 Since these are the same, we assign the value O to the third digit of ?±^sub 1^ : ?±^sub 1^ = 110x

4 Finally, to find the fourth digit of ?±^sub 1^ we examine the fourth and first digits of ?±^sub 0^ The fourth digit is 1 and the first is 1 Since these are the same, we assign the value O to the fourth digit of ?±^sub 1^ : ?±^sub 1^=1100

So we diocese that ?±^sub 0^ = 1011 generates the number ?±^sub 1^=1100 To check that the algorithm is understood, the reader may wish to verify that if ?±^sub 0^ = 0011101 then ?±^sub 1^ = 0100111 and if ?±^sub 0^ = 1111111110 then ?±^sub 1^ = 0000000011

Motivating Questions

1 A fundamental question is in what way the sequence {?±^sub k^ : k = 01 2} will behave for different sperms ?±^sub 0^. In other words, for which values of ?±^sub 0^ will the following terminate (i.e., become constant)? For which values of ?±^sub 0^ will the following become cyclical (i.e., periodic)? If the succession does terminate, how many iterations of the algorithm are required to guarantee termination? As we might wait for the answers to these questions be pendent on N, the length of ?±^sub 0^

Example 2:

a) Consider the se from Example 1 ?±^sub 0^=1011 In this case the succession {?±^sub k^ : k = 0 1 2 } terminates: ?±^sub 1^ -1100 ?±^sub 2^=0101 ?±^sub 3^=1111 ?±^sub 4^=000 ?±^sub 5^=0000 etc