Sequence Algebra - A Tutorial Paper
by
Huen Y.K.
CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg
(A short communication - 1st released: 2/2/98, Revised:3/2)
Abstract
Sequence algebra is familiar to workers in sampled-data control systems and
impluse signal analyses under the name of z-transform. Oddly enough z-transform
is more favoured by applied than pure mathematicians as evidenced
by the fact that most textbooks
on discrete mathematics give more emphasis on generating functions according to
the Euler's form [8,9]. The first indication of the power of z-transform outside its traditonal
domain came in 1994 when the author succeeded in deriving the global generating
function of the composite number sequence and subsequently the prime sequence
and the twin-prime sequence [1,2,3,4,7]. This was made possible when the author
proposed a break away from the Euler's form. Outside its traditional field, the author
has renamed it sequence algebra. From feedbacks, it is realised that papers hosted
in the URLsite so far are too
advanced for beginners. This paper is written in a tutorial style where the author
rearranges materials which have already appeared in earlier papers. The best way to
learn from these examples is that if you have access to a symbolic algebra package
such as
Maple V R 3 or Macsyma 2.2.
1. Introduction
This paper is written in a tutorial style with plenty of simple exercises which
you can follow if you have access to a symbolic algebra package such as Maple or
Macsyma. Even if you have not used symbolic softwares before, it should not
take you long to learn how to use it by simply doing cut and paste of various
program lines from this paper into your software and see them in actions. This is
because only a handful of supplied functions are used, mostly concerned with the
manipulations of Larent series using either Maple or Macsyma. Even if you are
new to symbolic algebra, you will find that it takes much less time to learn than
a compiled language. The program lines written in symbolic algebra would have
take ten to fifteen times more lines if written in compiled languages. The author
likes to use symbolic algebra in publications because program lines could be tested
easily by interested readers.
In order to understand the theory of sequence algebra, the first step is to understand
how sequences are represented in this algebraic format. Sequence algebra is the algebra
of global sequences which are manipulated algebraically in closed forms. Realistically,
for numerical work, one is confined to handling only finite sequences.
A group of objects
with common properties can be collected into a set. For example, equation (1) shows a finite
set of prime numbers.
Prime_set = [2,3,4,7,11,13,17,19,23]..........................................(1).
In the Euler's form, one might attempt to represent this sequence as shown in
equation (2).
..........................................2......3.........4.........5..........6.........7.........8
Prime_set := 2 + 3 x + 5 x + 7 x + 11 x + 13 x + 17 x + 19 x + 23 x ........(2).
If you are asked to find the closed form generating function for equation (2), you
will have great difficulty since the power-indices chosen to order this set of primes
assume that these numbers are placed in succession with no missing integers in
between them. This is one of the difficulties connected with the Euler's form. In sequence
algebra, the above prime set is represented as shown in equation (3).
Prime_set = [0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1] ....................(3).
Integers, whether primes or composites are no more represented by their magnitudes
but by their order in the natural number axis starting from the integer of zero. 1
represents the presence of an integer and 0 represents its absence. Thus the first integer
of 1 to appear is displaced from the first integer of zero by two unit
intervals and is thus understood to be the integer 2. Thus all the 1s in equation
(3) represent primes. This set would be meaningless if all the zeroes are removed
from it as is normally done in presenting a set of integers. It is surely unreadable
for a long sequence but one needs not do so using the Laurent series form as shown
in equation (4) where the power indices to the denominator order variables z indicate
the magnitudes of primes. The numerator coefficients indicate the duplicities of terms
and since in equation (4) these are all unity values, this means that there is only one
copy for each term, i.e., primes are unique and distinct values. In sequence algebra,
this is called a normalised sequence. Certain basic sequences such as Nat(z),
Even(z) and Odd(z) are intrinsic normalised sequences but higher sequences derived
from them could become unnormalised and can be reduced to the normalised
form using a newly defined function called Normc. These will be explained shortly.
Prime_set:=sum(1/z^i*(isprime(i)-false)/(true-false),i=0..23);
........................................1.......1.........1........1........1.......1.......1.......1......1
.................Prime_set := ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- ...................(4).
..........................................2.......3.........5........7.......11.....13.....17......19....23
........................................z........z.........z.........z........z........z.......z........z......z
A further advantage in sequence algebra is that most well known number sequences
such as Nat(z), Odd(z), Even(z) and so on have closed form in generating functions
which can be manipulated using symbolic algebraic packages.
For series expansions, symbolic packages traditionally offer both the power series
form and the Laurent series form. The Laurent series form is adopted because it offers some
advantges in algebraic manipulations. Each term of Laurent series expansion takes
the standard form as shown in equation
(5) where duplicity indicates the number of copies of each individual term and the power
index measures the number of unit internals from the first integer of zero. z is called an
order variable which is purely algebraic. The best analogy for order variables
is to view these like milestones
along a road marked by numbers indicating the distances relative to the
first milestone.
........................k.....................duplicity
...................... ---- = ---------------------------------.............................(5).
...........................i.................................power_index
........................z...........order_variable
The Laurent series form adopted in equation (5) turns out to be convenient since the
duplicity factors and the power-indices are visually partitioned into numerators and
denominators. This means that it is easy to split such a sequence into a
numerator and a denominator set using intrinsic functions such as ratnumer( ) and
denom( ) in Macsyma 2.2.
2. Tutorial Exercises
Tutorial examples are graded from simple to intermediate level. For best
benefit, please display this HTML page side-by-side with your symbolic algebra
package so that you can cut-and-paste program lines from the former to the latter
for interpretations. If you are not using either Maple or Macsyma, you may need
to modify the lines to suit. This is a problem peculiar to LISP which has
many dialects.
=================================================
Example 1: Generate a finite natural number set Nat(z) containing
7 contiguous integers
given by [1111111] in Laurent series form. In such a set, by convention, one starts
from the integer 0 and indicates magnitudes of integers by the number of intervals
from this point. For example, in Nat(z), "0" itself is an integer so the first element
should be written as 1 in the set instead of 0. Please
remember that 1 means presence and 0 means absence in the set.
Note: The term "contiguous" or "contiguity" has different meanings for different type of
number sequences.
For example a contiguous even number set is given by [1010101...] whereas a
contiguous odd number set is given by [01010101...]. So contiguity is a contextual
property which has meaning only if the specific number sequence is known.
Solutions:
Maple: Nat(z):=series(z/(z-1),z=infinity,7);
................................................1........1........1........1........1.............1
................Nat(z) := 1 + 1/z + ---- + ---- + ---- + ---- + ---- + O(----) .....................(E1a).
..................................................2........3........4........5........6.............7
................................................z........z.........z.........z........z.............z
Macsyma:fancy_display:false$
..................Nat_z:taylor(z/(z-1),z,inf,7);
.....................................1....1......1.....1.....1.....1.....1
(d4)/T/....................1 + - + -- + -- + -- + -- + -- + -- + . . . ....................................(E1b).
.....................................z.......2......3.....4.....5.....6.....7
............................................z......z.....z......z......z.....z
=================================================
Example 2: Prove that the sum of Even(z) and Odd(z) will generate
Nat(z).
Solutions:
.................................................................2
...............................................................z
...........................................Even(z) := ------..............................(E2a).
...............................................................2
.............................................................z - 1
................................................................z
............................................Odd(z) := ------..............................(E2b).
................................................................2
..............................................................z - 1
Maple:Nat(z):=factor(Even(z)+Odd(z));
Macsyma:Nat_z:factor(Even(z)+Odd(z));
.............................................................z
............................................Nat(z) := -----................................(E2c).
...........................................................z - 1
Note: By convention in sequence algebra, Nat(z) and Even(z) are defined
apriori as an axiom. An axiom or postulate cannot be proven. To avoid this circularity
trap, one should prove that Odd(z)=Nat(z)-Even(z)=z/(z^2-1) first. Then Nat(z)=
factor(Even(z)+Odd(z)) will be correct as given by equation (2c).
=================================================
Example 3: Nat(z) has two forms, viz., z/(z-1) and 1/(z-1). Also
Even(z) has two forms, viz., z^2/(z^2-1) and 1/(z^2-1). Explain the difference between
the two forms?
Solutions:
Given any generating function G(z), z^k*G(z) will shift the original sequence by
k unit intervals to the left whilst z^(-k)*G(z) will shift it to the right. These are best
explained using the two froms form Nat(z) and Even(z) given from equations (E3a) to
(E3d). Note that Nat(z)=z/(z-1) will start from the integer 1/z^0=1 whilst 1/(z-1)=z^(-1)*Nat(z)
start from the integer 1/z as shown in equations (E3a) to (E3b). The same remarks
apply to Even(z)=z^/(z^2-1) which starts from 1/z^0 whereas z^(-2)*Even(z) will start
from 1/z^0 as shown in equations (E3c) and (E3d).
Macsyma:Nat_1(z):taylor(z/(z-1),z,inf,7);
Maple:Nat_1(z):=series(z/(z-1),z=infinity,7);
...........................................1........1.........1........1........1.............1
........Nat_1(z) := 1 + 1/z + ---- + ---- + ---- + ---- + ---- + O(----).........(E3a).
.............................................2........3.........4........5........6............7
...........................................z........z..........z.........z........z.............z
Macsyma:Nat_2:=taylor(1/(z-1),z,inf,7);
Maple:Nat_2:=series(1/(z-1),z=infinity,7);
.......................................1........1........1..........1........1............1
..............Nat_2 := 1/z + ---- + ---- + ---- + ---- + ---- + O(----)..............(E3b).
.........................................2........3........4..........5........6...........7
.......................................z........z.........z...........z........z............z
Macsyma:Even_1:taylor(z^2/(z^2-1),z,inf,7);
Maple:Even_1:=series(z^2/(z^2-1),z=infinity,7);
..............................................1........1.........1............1
.....................Even_1 := 1 + ---- + ---- + ---- + O(----)................(E3c).
................................................2........4.........6............8
..............................................z........z.........z..............z
Macsyma:Even_2:taylor(1/(z^2-1),z,inf,7);
Maple:Even_2:=series(1/(z^2-1),z=infinity,7);
............................................1........1........1............1
.........................Even_2 := ---- + ---- + ---- + O(----)...................(E3d).
..............................................2........4........6.............8
............................................z.........z........z.............z
If Even(z)=z^2/(z^2-1), then Odd(z)=z^(-1)*Even(z) will shift the even integer sequence by one unit interval
to the right which makes it an odd integer sequence. Note that whereas Nat(z) and
Even(z) can be defined to start from the integer zero based on equation (E3a) and (E3c),
it is not possible to do the
same to Odd(z) once the former two sequences have been thus defined. In other words,
by definition, we have chosen to treat the integer zero as an even integer which will only
appear in Nat(z) and Even(z) but not Odd(z). This has profound effects on the properties
of Odd(z) which sets it apart from the former two sequences.
Macsyma:Odd_1:taylor(z/(z^2-1),z,inf,7);
Maple:Odd_1:=series(z/(z^2-1),z=infinity,7);
..................................................1.........1.............1
........................Odd_1 := 1/z + ---- + ---- + O(----)...................(E3e).
....................................................3.........5..............7
..................................................z..........z.............z
=================================================
Example 4:Fixed Deposit With Compound Interest: This example
is somewhat different from the usual applications of sequence algebra in number
theory. Suppose you
deposit a capital sum in a finance company at k% interest compounded annually.
How long does it take to double the initial captial if interest-rate r=6%, 9% and 12%
respectively. Note that sequence algebra can handle both integers and floating
point representations.
Solutions:
Macsyma: r:1.06;
Maple: r:=1.06;
Macsyma: Fixed_deposit:taylor(Initial_deposit*year/(year-r),year,inf,15);
Maple: Fixed_deposit:=series(Initial_deposit*year/(year-r),year=infinity,15);
.............................................................Initial_deposit................Initial_deposit
Fixed_deposit := Initial_deposit + 1.06 --------------- + 1.1236 ---------------
.......................................................................year....................................2
..........................................................................................................year
...................Initial_deposit...........................Initial_deposit...........................Initial_deposit
+ 1.191016 --------------- + 1.26247696 --------------- + 1.338225578 ---------------
..................................3..............................................4..........................................5
...........................year.........................................year.....................................year
.........................Initial_deposit............................Initial_deposit..............................Initial_deposit
+ 1.418519113 --------------- + 1.503630260 --------------- + 1.593848076 ---------------
.........................................6..............................................7..............................................8
..................................year.........................................year.........................................year
.........................Initial_deposit.............................Initial_deposit.............................Initial_deposit
+ 1.689478961 --------------- + 1.790847699 --------------- + 1.898298561 ---------------
.........................................9.............................................10.............................................11
..................................year.........................................year..........................................year
..........................Initial_deposit............................Initial_deposit.............................Initial_deposit
+ 2.012196475 --------------- + 2.132928264 --------------- + 2.260903960 ---------------
.........................................12...........................................13............................................14
...................................year........................................year..........................................year
............1
+ O(------)..........................................(E4a).
..............15
.........year
r:=1.09;
Fixed_deposit :=
+..................................+............
......................... Initial_deposit.............................Initial_deposit..........................Initial_deposit
+ 1.677100111 --------------- + 1.828039121 --------------- + 1.992562642 ---------------
..........................................6..............................................7..................................................8
...................................year.........................................year...........................................year
+..................................+........... .............................(4b).
r:=1.12;
Fixed_deposit :=
+.......................................+................
.........................Initial_deposit...........................Initial_deposit...............................Initial_deposit
+ 1.973822685 --------------- + 2.210681407 --------------- + 2.475963176 ---------------
.........................................6...............................................7...................................................8
.................................year.........................................year...............................................year
+......................................+.......... ...........................(E4c).
Results: It can be seen that the capital doubling period for the three interest
rate r are given as follows:
r=6%: 12 years
r=9%: slightly more than 8 years
r=12%: slightly less than 6 years.
In sequence algebra the standard form for the denominator is to have r = 1
which can be obtained by dividing both numerator and denominator by r as shown
in equation (E4d).
...........................................z..............z
.........................................----- = ----------- .......................................(E4d).
.........................................z - r.......r (z/r - 1)
=================================================
Example 5a: Linear Sequence Product: Find the linear sequence product
of two identical number sequences given by:
................................................a(1)....a(2)....a(3)....a(4)....a(5)
..........................F(z) := a(0) + ---- + ---- + ---- + ---- + ---- ......................(E5a).
..................................................z...........2........3........4........5
............................................................z.........z........z.........z
Comments:Originally the author called this
inner product but in symbolic packages, this term applies to functions either in Lie
algebra or matrices. Linear sequence product is the term by term product
between two identically ordered sequences with the same number of terms.
This is the opposite of outer sequence product given in example 6. At present the
function inner sequence product is not provided by symbolic packages. Also note
that the two sequences only need to be identically ordered but can have
nonidentical numerator values. This is demonstrated in example 5b.
Solutions:
.........................................................2........2........2........2.........2
...............................................2...a(1)....a(2)....a(3)....a(4)....a(5)
linear_seqprod(F(z)^2) := a(0) + ---- + ---- + ---- + ---- + ---- ......................(E5b).
.......................................................z...........2........3........4........5
.................................................................z.........z........z.........z
Example 5b: Find the linear sequence product between y1 and y2 where these two
sequences are identically ordered and contain the same number of terms.
fancy_display:false$
x1:sum(a(i)/z^i,i,0,5);
..................a(1).....a(2)...a(3)...a(4)....a(5)
..................---- + ---- + ---- + ---- + ---- .........................................(E5c).
.....................z.........2.........3.......4.........5
..............................z.........z.......z.........z
Num_set1(x1):=block([args:args(x1)],apply("[",args*map('denom,args)))$
or
Num_set1(x1):=block([args:args(x1)],args*map('denom,args))$
xxx:Num_set1(x1);
............................[a(1), a(2), a(3), a(4), a(5)].........................(E5d).
y1:sum(b(i)/z^i,i,1,5);
...........................b(1).....b(2)...b(3)...b(4)....b(5)
...........................---- + ---- + ---- + ---- + ----..............................(E5e).
.............................z...........2........3........4.........5
........................................z........z.........z.........z
Num_set2(y1):=block([args:args(x1)],apply("[",args*map('denom,args)))$
or
Num_set2(x1):=block([args:args(x1)],args*map('denom,args))$
yyy:Num_set2(y1);
...........................[b(1), b(2), b(3), b(4), b(5)]
y3:makelist(pop(xxx)*pop(yyy)/z^i,i,1,5);
In the above program-line we use pop function which is more familiar to machine
code programmers. Given a list [x1,x2,x3,...], each pop operation applied to this
list will pop one item from the head of the list. We will get successively, x1, x2,...
being popped out and the original list will finally become empty. Push is the opposite
of pop and can be used to restore to the original list.
..........................................a(1) b(1)..a(2) b(2)..a(3) b(3)..a(4) b(4)..a(5) b(5)
........................................[ ---------, ---------, ---------, ---------, ---------]..........(E5e).
................................................z...............2..............3..............4..............5
...............................................................z..............z..............z..............z
Linear_seqproduct:apply("+",y3);
The result is not a list, hence it is not enclosed in square brackets. A single
atom cannot form a list.
.....................................a(1) b(1)...a(2) b(2)...a(3) b(3)....a(4) b(4)....a(5) b(5)
....................................--------- + --------- + --------- + --------- + ---------.........(E5f).
...........................................z................2................3................4................5
...........................................................z................z.................z................z
====================================================
Example 6: Given the list [0,1,0,1,0,1,0,1,0,1], reassemble this back
into sequence algebraic format.
Comments: This is relatively straightforward compared to that of example 5.
A set within square bracket in symbolic algebra is a list. Only
program lines in Macsyma are displayed for the same reason as given in the previous
example.
Solutions:
coeffs:[0,1,0,1,0,1,0,1,0,1]; ........................................(E6a).
makelist(Coeffs[i]/z^(i-1),i,1,length(Coeffs));
............................1........1........1.........1........1
.....................[0, ---, 0, ---, 0, ---, 0, ---, 0, --- ]..................................(E6b).
............................z..........3.........5.........7........9
......................................z.........z..........z........z
apply("+",%);
....................1.......1.......1......1.......1
...................--- + --- + --- + --- + --- .................................................(E6c).
....................z.........3........5......7.......9
.............................z........z.......z.......z
===================================================
Example 7: Given the algebraic sequence in equation (7a), find the
integer-set as well as the power-index set. [13]
Solutions:
Only
program lines in Macsyma are displayed for the same reason as given in the previous
example.
Nat_z:1+1/z+1/z^3+1/z^5+1/z^6;
.......1......1.......1.......1
.....--- + --- + --- + --- + 1 ..................................................................(E7a).
.......z........3.......5.......6
...............z.......z.......z
Integer_set:reverse(makelist(coeff(Nat_z,z,i),i,lopow(Nat_z,z),0));
.......................[1, 1, 0, 1, 0, 1, 1]...................................................(E7b).
Power_index:reverse(makelist(abs(i),i,lopow(Nat_z,z),0));
.......................[0, 1, 2, 3, 4, 5, 6]...................................................(E7c).
Comments on solutions: The abs( ) function is inserted in equations (E7a, E7b)
since the natural number set is positive. Without it the power indices will be returned
as negative since these are in the denominators in equation (E7a). Note that the standard
format for presenting an integer number set is by equation (E7b) and not (E7c). Equation
(E7c) will always return the natural number set since in sequence algebra, all number
sequences are considered as subsets of the natural number sequence.
=========================================================
Example 8: Cross sequence product between two sequences: This operation
is conventional and is normally just referred to as sequence product. We call this
by the present name to distinguish it from linear sequence product. This operation
is provided by all symbolic packages.
Given two sequences F1(z) and F2(z), find its cross sequence product.
Solutions:
..............................................................a(1)....a(2)
......................................F1(z) := a(0) + ---- + ---- ..........................(E8a).
................................................................z...........2
..........................................................................z
..........................................................b(1)....b(2)....b(3)
..................................F2(z) := b(0) + ---- + ---- + ---- .....................(E8b).
............................................................z..........2.........3
......................................................................z.........z
Macsyma: Cross_prod(z):sort(expand(F1(z)*F2(z)));
Maple: Cross_prod(z):=sort(expand(F1(z)*F2(z)));
............................a(1) b(0)...a(0) b(1)....a(0) b(2)....a(2) b(0)...a(1) b(1)....a(0) b(3)....a(1) b(2)
Cross_prod(z) := --------- + --------- + --------- + --------- + --------- + --------- + ---------
..................................z................z.................2................2................2...............3................3
....................................................................z................z.................z...............z................z
....a(2) b(1)....a(1) b(3)....a(2) b(2)....a(2) b(3)
+ --------- + --------- + --------- + --------- + a(0) b(0) ...........................(E8c).
.........3...................4................4.................5
.......z...................z................z.................z
With 3 terms in equation (E8a) and 4 terms in equation (E8b), we expect the resultant
cross sequence product to have 12 terms. Also if all array elements are unity valued,
one should expect the distribution to take a triangular form as: 1-2-3-3-2-1. In other
words, after collecting like ordered terms, there should be only 6 different terms. This
formation is general with contiuguous ordered terms in that the largest duplicity always
occur at mid-sequence range. However if the two input sequences are differently
ordered with gaps between terms, this rule should be applied with care.
=================================================
Example 9: Chart Of Primitive And Secondary Number Sequences [4]: In sequence
algebra, primitive sequences are those which exists as normalised number sequences
such as Nat(z), Even(z) and Odd(z). Beyond these are are sequences derived from
primitive sequences and are called secondary sequences. These sequences are
always returned unnormalised but can be normalised using the Normc function. For
an exercise, use a symbolic package to derive Oddcomp(z) and Prime(z). Note that
the derivation of Comp(z) is more involved but has already been covered by previous
papers. Here Comp(z) will be supplied for the problem.
Solutions: We start with the axiomatic defintions of Nat(z) and Even(z) given by equations (E9a)
and (E9b).
...............................................................z
............................................Nat(z) := ----- ...........................(E9a).
.............................................................z - 1
..................................................................2
................................................................z
..........................................Even(z) := ------...........................(E9b).
................................................................2
..............................................................z - 1
From the above, we can derive Odd(z) in equation (E9c):
Macsyma: Odd(z):factor(Nat(z)-Even(z));
Maple: Odd(z):=factor(Nat(z)-Even(z));
..............................................................z...................z
....................................Odd(z) := --------------- = ------ ..........................(E9c).
..................................................................................2
.....................................................(z - 1) (z + 1).....(z - 1)
We also supply the equation of divisibles or Comp(z) given by equation (9d):
...................................................................1
.......................................Comp(z) := ----------- ...................................(E9d).
.................................................................i...i
..............................................................z (z - 1)
Comp(z):=series(sum(1/(z^i*(z^i-1)),i=2..20),z=infinity,20);
....................1.........2........2........1.......2.......4.......2.......2.......3.......4..........1
Comp(z) := ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + O(---) ..........(9e).
......................4.........6........8........9......10.....12.....14.....15.....16.....18........20
....................z.........z.........z.........z.......z.......z........z........z.......z........z..........z
..................................1........1........1........1.......1.......1.......1........1......1........1
Normc(Comp(z)) := ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- ...........(E9f).
....................................4........6........8........9......10......12.....14.....15.....16.....18
..................................z........z.........z........z........z........z.......z........z........z........z
The program lines assume that the function Normc( ) is already available.
Macsyma: Oddcomp(z):Normc(Comp(z))-Even(z)+1+1/z^2;
Maple: Oddcomp(z):=Normc(Comp(z))-Even(z)+1+1/z^2;
............................................1........1...........1
..................Oddcomp(z) := ---- + --- + O(---) ........................................(E9g).
..............................................9.......15.........20
............................................z.........z...........z
Macsyma: Prime(z):Nat(z)-Comp(z)-1-1/z-1/z^2;
Maple: Prime(z):=Nat(z)-Comp(z)-1-1/z-1/z^2;
....................1.......1........1.......1.......1........1.......1...........1
Prime(z) := ---- + ---- + ---- + --- + --- + --- + --- + O(---) .......................(E9h).
......................3.......5........7.......11.....13.....17......19........20
....................z.......z.........z........z.......z........z.........z..........z
An important family chart of primitive and secondary number sequences are
given in figure 1 which first appeared in a recently published paper [4].
..................................................Nat(z)=z/(z-1)
............................................................|
........................------------------------------------------------
........................|...................................|...................................|
....................Even(z)........................Comp(z)......................Odd(z)
....................=z^2/(z^2-1)..............=1/(z^i*(z^i-1)).............=z/(z^2-1)
........................|...................................|...................................|
........................------------------------...................................|
...........................................|....................................................|
...................................Oddcomp(z).........................................|
..................=Normc(Comp(z)-Even(z)+1+1/z^2......................|
...........................................|....................................................|
...........................................------------------------------------
....................................................................|
..............................................................Prime(z)
....................................................=Odd(z)-Oddcomp(z)
..............Fig. 1 - Family chart of primitive and secondary sequences.
3. Summary
This tutorial paper demonstrates by examples how number sequences are
represented and manipulated in sequence algebra. Certain operations which
require advanced programming methods using symbolic packages have not
been included. More papers will be added on symbolic programming relevant
to requirments in sequence algebra in future. This is because there is a strong
synergistic bond between sequence algebra and symbolic algebra softwares
which leads to the discoveries of new mathematical operations especially in
domains which are at present not heavily influenced by mathematics such as
life sciences. Although sequence algebra originally finds it niche in holistic
number theory, it originated from z-transform which should be regarded as
a branch of applied mathematics. This is because z-transform was developed
to its present level by applied scientists rather than by pure mathematicians [10].
4. References
Comments: Not all references in this list are directly referred in the main paper.
Most are provided for readers not familiar with sequence algebra. These papers
can be easily hyperlinked whilst you are in the web.
1. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci.
Technol., 1994, VOL. 25, NO.6, pp 913-920.
2. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ.
Sci. Technol., 1996, VOL.27, NO. 5, 685-691.
3. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,
1997, VOL.28, NO.3, pp 333-344.
4. Huen Y.K.: The twin prime problem revisited, INT. J. Math. Educ. Sci. Technol.,1997, VOL.28,
NO. 6, pp 825-834.
5. Huen Y.K.: Is Pie Periodic?, INT. J. Math. Educ. Sci. Technol.,199?,VOL.??,NO.?,???-???. (in the press).
6. Huen Y.K.: Final value theorem in number sequences., INT. J. Math. Educ. Sci. Technol.,199?,VOL.-??,NO.?,???-???. (accepted).
7. Burton M. Burton: Elementary Number Theory, Third Edition, WCB publishers, 1994, 17 to
48 to 50.
8. Brualdi A. Richard (1977): Introductory Combinatorics, Chapter 7, Generating
Functions, North Holland, pp 127 to 139.
9. Liu C.L. (1985): Elements Of Discrete Mathematics, McGRAW-HILL
INTERNATIONAL EDITIONS, computer science series, pp 290 to 295.
10. Luyben W.L. : Process Modeling, Simulation And Control For Chemical
Engineers,1990 (second edition), International Edition, MacGraw-Hill Publishing
Company, pp 402 to 404.
11. A Simple Introduction To Sequence
Algebra - by Huen Y.K.
(date release: 15.3.97) (38 KBytes, 11*A4 pages).
========================================================
12. Evaluations Of Normc( ) Function
In Macsyma 2.2
- Huen Y.K. (Date Released 17/12/97, 14 Kbytes)
================================================
13.
List Processing In Sequence Algebra
- Huen Y.K. (Date Released 23/12/97, 20 Kbytes)
================================================
14. The Canonical Generating Function
or CGF(z) ... - by Huen Y.K.
(date released : 27.5..97) (24 KBytes, 7*A4s).
========================================================
15. Visual Solutions Of Number Theoretic
Problems ..... - by Huen Y.K. (date released : 3.6.97) (38.3 KBytes, 10*A4s).
========================================================
16. Final Value Theorem Applied To Number
Sequences... - by Huen Y.K. (date released : 5.6.97) (29.4 KBytes, 9*A4s).
========================================================
17. Unsolved Problems In Sequence
Algebra - by Huen Y.K. (date released : 6.6.97) (29.4 KBytes, 9*A4s).
========================================================
18. Methods Of Developing Sequence
Algebraic Formulations For Comp(z) and Prime(z) - by Huen Y.K. (date released : 20.6.97) (36.8 KBytes, 10*A4s).
========================================================
19. Composite Number Sequence
Challenge 1/97 - by Huen Y.K. (date released : 28.6.97) (24.8 KBytes, 7*A4s).
========================================================
20. Lemmata, Corollaries, And
Theorems In Sequence Order Analysis. - by Huen Y.K. (date released : 6.7.97) (38.3 KBytes, 12*A4s).
========================================================
21. Improved Formulations For Comp(z)
and Prime(z)
- by Huen Y.K. (date released : 16.9.97) (17 KBytes ).
========================================================
22. Detecting False Reports
in Primality Tests By The Oddcomp(z) Method.
- by Huen Y.K. (date released : 18.9.97, Revised 20/9) (26 KBytes ).
========================================================
23. The Throwing Power Of
Oddcomp(z).
- by Huen Y.K. (date released : 24.9.97 ) (15 Kbytes).
========================================================
24. Sequence Algebraic
Approach To Prime Number Theorem
- by Huen Y.K. (date released : 28.9.97 ) (21 Kbytes).
========================================================
25. Generating Functions -
Closed Forms vs Open Forms
- by Huen Y.K. (date released : 1.10.97 ) (21 Kbytes).
========================================================
26. Generating Large
Odd Composite With Two Prime Factors
- by Huen Y.K. (date released : 3.10.97 ) (13.5 Kbytes).
========================================================
27. In Search Of Counter-
Examples In Maple's Isprime Function.
- by Huen Y.K. (date released : 4.10.97 ) (18 Kbytes).
========================================================
28. A Sequence Algebraist's
View Of Lehmann's Primality Test
- by Huen Y.K. (date released : 6.10.97 ) (26 Kbytes).
========================================================
29. On Odd(z), Oddcomp(z),
Seq1(z) and Seq2(z)
- by Huen Y.K. (date released : 10.10.97 ) (17 Kbytes).
========================================================
30. How To Generate A Short
And Contiguous Oddcomp(z) Sequence?
- by Huen Y.K. (date released : 15.10.97 ) (13 Kbytes).
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