ON INTEGER SEQUENCES

ON INTEGER SEQUENCES
Keneth Adrian P. Dagal
Department of Mathematics
Ateneo de Naga University
February 4, 2012
INTRODUCTION
What are the things that come into our mind when we hear the word
sequence?
Perhaps, these are some overt words that most of us automatically think
of: order, arrangement, pattern, rules, and arithmetic sequence. Yes, these
are indeed very conventional. Now, we will use some concepts on
functions, set theory, algebra, and many more, to make an immense view
of sequences.
SEQUENCE
A sequence is a serial arrangement in which things follow in logical order or a recurrent
pattern(Wordweb 6.72 ). From this definition, it is clear that sequences are
arrangements that are accompanied with logical order or a recurrent pattern. What is
very much interesting about this is that: Suppose we are given a certain arrangement
and then we are asked to give the next entry or the formula that will generate all the
values in the sequence. This is quite evident when we try to devise formulas in
algebra, trigonometry, geometry, combinatorics, and some basic math concepts. It
should be noted though that in a given arrangement, it is of great difficulty to claim
that there is no pattern or it is chaotic. The concept of order is perplexed and
modifications are necessary for clarity.
ILLUSTRATION
Before we tackle the axiomatic approach of sequences, we will first
have some variety of problems in which sequences are of great
importance. The problems may be non –routine problems, but still
give us how our students work on the problem. We are fond of
generalizing and making formulas so that our calculations are
minimized and we are able to optimize the time and the
computations.
PROBLEMS
PROBLEMS
PROBLEMS
PROBLEMS
SOLUTIONS
CLAIM 1
PROOF
CLAIM 2
PROOF
CLAIM 3
PROOF
PROOF
CLAIM 4
PROOF
Size
Number of squares
1x1 square
2x2 square
3x3 square
.
.
.
.
.
.
1
PROOF
CIRCULAR SEQUENCE
CLAIM 5
PROOF
CLAIM 6
PROOF
SOME NOTES
DERANGEMENT
DEMONTMORT
SIPIS FUNCTION
PROPERTIES
PROPERTIES
NUMEROSITY
FIBONNACI SEQUENCE
WELL-KNOWN SEQUENCES
ARITHMETIC SEQUENCE
DIFFERENCE ALGORITHM
The algorithm is one way of telling if a given sequence has a
polynomial closed form. But it doesn’t guarantee us that all
polynomial sequences have this same behavior; that require an
extensive proof. If the algorithm does not halt, it implies that the
given sequence is not of a polynomial sequence.
We consider the reverse, that is instead of having random sequences
and finding its closed form, if there is, we try to start reversing the
algorithm and generate sequences with varying polynomial closed
form.
THEOREM
PROOF (AN)
A computer program could be of great importance when generating
such sequence. A closed form for the sequence will be the input.
Now, I have devised a way in finding such closed form after doing the
difference algorithm. A background in calculus is important. The
methodology is just a way in finding such closed form but it is not yet
proven for its certainty, a minor revision is quite a need in the
generalization of the concept.
METHODOLOGY
ILLUSTRATION
In the investigation of the solution, it may be good to show its
validity or not. And explain further why it works in some cases.
Limiting the usage is the way to fully understand the concept
presented.
CONCLUSION
 Sequences are ineluctable in studying mathematics. They are hidden in
a lot of mathematical ideas most especially in counting problems. It is
actually an important factor in honing such mathematical skills. What I
have just discussed is a mere portion of the vast scope of sequences. My
concern isn’t about convergence since the sequences I consider are all
divergent. Another thing is, this work helps student in contriving formulas
and proving such. And I only include sequences to be SIPIS that makes a
lot of limitations but the treatment is quite extensive and deep.
THANK YOU