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Abstract
Objectives: The paper aims to present a survey of both time-honored and contemporary studies on triangular number,
factorial, relationship between the two, and some other numbers associated with them. Methods: The research is
expository in nature. It focuses on expositions regarding the triangular number, its multiplicative analog – the factorial
and other numbers related to them. Findings: Much had been studied about triangular numbers, factorials and other
numbers involving sums of triangular numbers or sums of factorials. However, it seems that nobody had explored the
properties of the sums of corresponding factorials and triangular numbers. Hence, explorations on these integers,
called factoriangular numbers, were conducted. Series of experimental mathematics resulted to the characterization
of factoriangular numbers as to its parity, compositeness, number and sum of positive divisors and other minor
characteristics. It was also found that every factoriangular number has a runsum representation of length k, the first
term of which is
( 1)! 1k−+
and the last term is
( 1)! kk−+
. The sequence of factoriangular numbers is a recurring
sequence and it has a rational closed-form of exponential generating function. These numbers were also characterized as
to when a factoriangular number can be expressed as a sum of two triangular numbers and/or as a sum of two squares.
Application/ Improvement: The introduction of factoriangular number and expositions on this type of number is a
novel contribution to the theory of numbers. Surveys, expositions and explorations on existing studies may continue to be
a major undertaking in number theory.
A Survey on Triangular Number, Factorial and Some
Associated Numbers
Romer C. Castillo
College of Accountancy, Business, Economics and International Hospitality Management, Batangas State
University, Batangas City – 4200., Philippines;romercastillo@rocketmail.com
Keywords: Factorial, Factorial-like number, Factoriangular number, Polygonal number, Triangular number
1. Introduction
In the mathematical eld, a sense of beauty seems to be
almost the only useful drive for discovery1 and it is imag-
ination and not reasoning that seems to be the moving
power for invention in mathematics2. It is in number the-
ory that many of the greatest mathematicians in history
had tried their hand3 paving the way for mathematical
experimentations, explorations and discoveries. Gauss
once said that the theory of numbers is the queen of
mathematics and mathematics is the queen of science4.
e theory of numbers concerns the characteristics
of integers and rational numbers beyond the ordinary
arithmetic computations. Because of its unquestioned
historical importance, this theory had occupied a central
position in the world of both ancient and contemporary
mathematics.
As far back as ancient Greece, mathematicians were
studying number theory. e Pythagoreans were very
much interested in the somewhat mythical properties
of integers. ey initiated the study of perfect numbers,
decient and abundant numbers, amicable numbers,
polygonal numbers, and Pythagorean triples. Since then,
almost every major civilization had produced number
theorists who discovered new and fascinating properties
of numbers for nearly every century.
Until today, number theory has shown its irresistible
appeal to professional, as well as beginning, mathemati-
cians. One reason for this lies in the basic nature of its
problems5. Although many of the number theory prob-
lems are extremely dicult to solve and remain to be the
*Author for correspondence
Indian Journal of Science and Technology, Vol 9(41), DOI: 10.17485/ijst/2016/v9i41/85182, November 2016
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
A Survey on Triangular Number, Factorial and Some Associated Numbers
Indian Journal of Science and Technology
2Vol 9 (41) | November 2016 | www.indjst.org
most elusive unsolved problems in mathematics, they can
be formulated in terms that are simple enough to arouse
the interest and curiosity of even those without much
mathematical training.
More than in any part of mathematics, the methods of
inquiry in number theory adhere to the scientic approach.
ose working on the eld must rely to a large extent
upon trial and error and their own curiosity, intuition
and ingenuity. Rigorous mathematical proofs are oen
preceded by patient and time-consuming mathematical
experimentation or experimental mathematics.
Experimental mathematics refers to an approach of
studying mathematics where a eld can be eectively
studied using advanced computing technology such as
computer algebra systems6. However, computer system
alone is not enough to solve problems; human intuition and
insight still play a vital role in successfully leading the math-
ematical explorer on the path of discovery. And besides,
number theorists in the past have done these mathematical
experimentations only by hand analysis and computations.
Among the many works in number theory that can be
done through experimental mathematics, exploring pat-
terns in integer sequences is one of the most interesting and
frequently conducted. It is quite dicult now to count the
number of studies on Fibonacci sequence, Lucas sequence,
the Pell and associated Pell sequences, and other well-known
sequences. Classical number patterns like the triangular
and other polygonal and gurate numbers have also been
studied from the ancient up to the modern times by mathe-
maticians, professionals and amateurs alike, in almost every
part of the world. e multiplicative analog of the triangular
number, the factorial, has also a special place in the litera-
ture being very useful not only in number theory but also
in other mathematical disciplines like combinatorial and
mathematical analysis. Quite recently, sequences of integers
generated by summing the digits7 were also being studied.
Integer patterns or sequences can be described by
algebraic formulas, recurrences and identities. As an
instance, the triangular number ( k
T
), for k ≥ 1, is deter-
mined by
( 1) / 2
k
T kk=+
. Given
k
T
the next term in
the triangular number sequence is also determined by
recurrence:
11
kk
T Tk
+
= ++
.
Some identities on triangular numbers are also
given in the literature, for instance8-10: 2
1
;
kk
TT k
−
+=
2
1( 1) ;
kk
TT k
+
+ =+
22 3
1( 1) ;
kk
TT k
+− =+ 1
( 2) ;
kk
kT k T
+
=+
2
22
1 ;
kk
k
T TT
−
=+
21
3 ;
k kk
T TT
−
=+
2
2
2 ;
kk
T Tk−=
and
2
8 1 (2 1)
k
Tk
+= + .
ere is also an identity involving the triangular
number and factorial. is relationship between factorials
and triangular numbers is given by11,12:
21
1
(2 )! 2
n
n
k
k
nT
−
=
=
∏
. Aside from this, there is a somewhat natural relation on
these factorials and triangular numbers: the triangular
number is regarded as the additive analog of factorial.
e current work aims to present a survey of both
time-honored and contemporary studies on triangu-
lar number and other polygonal numbers, factorial and
factorial-like numbers, and some other related or associ-
ated numbers. It also includes some interesting results of
recent studies conducted by the author regarding the sum
of corresponding factorial and triangular number, which
is named as factoriangular number.
2. Survey on Triangular Number,
Factorial and Related Numbers
e history of mathematics in general and the history
of number theory in particular are inseparable. Number
theory is one of the oldest elds in mathematics and most
of the greatest mathematicians contributed for its devel-
opment3. Although it is probable that the ancient Greek
mathematicians were largely indebted to the Babylonians
and Egyptians for a core of information about the proper-
ties of natural numbers, the rst rudiments of an actual
theory are generally credited to Pythagoras and his
followers, the Pythagoreans5.
2.1 Triangular Number and Other
Polygonal Numbers
An important subset of natural numbers in ancient Greece
is the set of polygonal numbers. e name polygonal
number was introduced by Heysicles to refer to positive
integers that are triangular, oblong, square, and the like13.
ese numbers can be considered as the ancient link
between number theory and geometry.
e characteristics of these polygonal numbers
were studied by Pythagoras and the Pythagoreans. ey
depicted these numbers as regular arrangements of dots
in geometric patterns. e triangular numbers were rep-
resented as triangular array of dots, the oblong numbers
as rectangular array of dots, and the square numbers as
square array of dots. ey also found that an oblong num-
ber is a sum of even numbers; a triangular number is a
sum of positive integers; and a square number is a sum
of odd numbers13. In modern notation, if the triangular
Romer C. Castillo
Indian Journal of Science and Technology 3
Vol 9 (41) | November 2016 | www.indjst.org
ranging from simple relationships between them and the
other polygonal numbers to very complex relationships
involving partitions, modular forms and combinatorial
properties8. Many other important results on these num-
bers have been discussed in the literature. A theorem of
Fermat states that a positive integer can be expressed as a
sum of at most three triangular, four square, ve pentago-
nal, or n n-gonal numbers. Gauss proved the triangular
case10,15. Euler le important results regarding this theo-
rem, which were utilized by Lagrange to prove the case
for squares and Jacobi also proved this independently16.
Cauchy showed the full proof of Fermat’s theorem15,16.
e expression of an integer as a sum of three
triangular numbers can be done in more than one way
and Dirichlet showed how to derive such number of
ways10. e modular form theory can also be used to cal-
culate the representations of integers as sums of triangular
numbers17. e ways a positive integer can be expressed as
a sum of two n-sided regular gurate numbers can also be
generalized18. Furthermore, it was shown that a generating
function manipulation and a combinatorial argument can
be used on the partitions of an integer into three triangu-
lar numbers and into three distinct triangular numbers,
respectively19.
e theory of theta functions can be used also to
compute the number of ways a natural number can be
expressed as a sum of three squares or of three triangular
numbers20. Following that, several studies on the mixed
sums of triangular and square numbers were conducted.
Any positive integer was shown to be a sum of two
triangular numbers and an even square and that each nat-
ural number is equal to a triangular number plus
22
xy
+
with x not congruent to y modulo 2 and where x and y are
integers21. ree conjectures on mixed sums of triangular
and square numbers were veried21, the rst for
15000n≤
while the second and third for
10000n≤
. e second
conjecture was later proven by using Gauss-Legendre
eorem and Jacobi’s identity22. e rst conjecture had
been later proven23 as well while the generalized Riemann
hypothesis implied the third conjecture about which
explicit natural numbers may be represented15.
Another conjecture states that for non-negative integers
m and n, every suciently large positive integer can be
expressed as either of the following: (1)
22
22
mn
z
x yT++
, (2)
2
22
mn
yz
x TT++
, (3)
22
mn
x yz
T TT++
, (4)
22
23
n
z
x yT+⋅ +
,
(5)
223
n
yz
x TT+⋅ +
, (6)
2
23 2
n
yz
x TT⋅++
, (7) 23 2
n
x yz
T TT⋅+ +
,
(8) 25
n
xyz
TTT⋅ ++
, (9)
234
xyz
TTT++
, (10) 22
232
z
xyT++
.
Six of the ten forms: (1), (4), (5), (6), (9) and (10), were
number is denoted by k
T
, the oblong number by k
O
, and
the square number by
k
S
, then
1
1 2 3 ...
k
k
i
Ti k
=
= =+ ++ +
∑ ;
1
2 2 4 6 ... 2
k
k
i
Oi k
=
= =+++ +
∑; and
1
(2 1) 1 3 5 . .. ( 2 1)
k
k
i
Si k
=
= − =+++ + −
∑.
It is very evident that
2kk
TO=
. Relating triangular
and square numbers, Nicomachus found that
2
11
( 1)
kk k
TT S k
++
+ = =+or
2
1k kk
T TSk
−
+= =
. He also
noted that triangular numbers can be produced from
square and oblong numbers13 in particular,
2kk k
SOT+=
and
1 21kk k
OS T
++
+=
. In addition, Plutarch also found13
that 2
21
8 1 (2 1)
kk
TS k
+
+= = + .
Further, Nicomachus proved13 that, if the pentagonal
number is denoted by 5
k
P, then for
1k>
, 5
1
3
kk
P Tk
−
=+
and 5
1
k kk
P ST
−
=+ .
If the polygonal number is be denoted by n
k
P, where
n is the number of sides, then the triangular number
can be denoted by 3
k
P and the square number by 4
k
P.
Using these new notations, the relations among pentag-
onal, square and triangular numbers is now given, for
1k>
, by
5 43
1k kk
PPP
−
=+
. Nicomachus generalized this
and claimed that, for
1k>
and
3n≥
,
13
1
nn
k kk
PP P
−
−
=+
holds for every n-gonal number14. is shows that every
polygonal number can be generated using the triangular
numbers.
For instance, squares can be generated using
1k kk
S TT
−
=+
, as Nicomachus observed; pentagonal num-
bers using
5
11
2
k kk k k
P ST T T
−−
=+ =+
; hexagonal numbers
using
65
11
3
k kk k k
P PT TT
−−
= + =+
; and in general, n-go-
nal numbers using 1
11
( 3)
nn
k kk k k
P P T TnT
−
−−
= + =+− ,
which is used to establish the closed formula for any kth
polygonal number14, for
1k≥
and
3n≥
,
( 2 4)
2
n
k
k kn k n
P
− −+
=.
Other established relations between triangular and
other polygonal numbers are as follows10: 6
21kk
PT
−
= and
5
1
31
3
kk
PT
−
=.
2.2 Contemporary Studies on Triangular
Numbers
Triangular numbers, though having a simple denition,
are amazingly rich in properties of various kinds,
A Survey on Triangular Number, Factorial and Some Associated Numbers
Indian Journal of Science and Technology
4Vol 9 (41) | November 2016 | www.indjst.org
proven while for the four other forms: (2), (3), (7) and (8),
counterexamples were found15.
2.3 Square Triangular Number and Related
Numbers
Square triangular numbers or numbers that are both
square and triangular are also well-studied. A square tri-
angular number can be written as n2 for some n and as
k(k+1)/2 for some k, and hence, is given by the equation
2
( 1)
2
kk
n
+
=
which is a Diophantine equation for which integer
solutions are demanded. Solving this equation by algebraic
manipulations leads to
22
(2 1) 2(2 ) 1kn+− =
and by change of
variables
21ak=+
and
2bn=
becomes
22
21ab−=
, which
is factorable into
( 2)( 2) 1
a ba b
+ −=
in nding integer solu-
tions. In a more general equation
22
1a kb−=
where k is a
xed integer, if
(,)ab
is a solution then
2
( / 2)kb=
is a square
triangular number24.
e sequence of square triangular numbers is
{1, 36, 1225, 41616, 1413721, 48024900, 1631432881,
55420693056, …}. e kth square triangular number,
which can be denoted by k
St
, can be obtained from the
recursive formula,
12
34 2
k kk
St St St
−−
= −+
for
3k≥
with
11St =
and
236St =
. e non-recursive formula, for
1k≥
,
( ) ( )
2
22
12 12
42
kk
k
St
+ −−
=
also gives the kth square triangular number25.
Square triangular numbers are related to
balancing numbers. A balancing number is
a positive integer k that makes the equation
1 2 ... ( 1) ( 1) ( 2) ... ( )k k k kr+++−=++++++
true for a
positive integer r called balancer26. e equation is equiv-
alent to
1k kr k
TTT
−+
=−
or
1k k kr
T TT
−+
+=
, which is also the
same as 2( )(( 1) / 2k krkr= + ++ . It is clear that when k2
is a triangular number, k is a balancing number. Hence,
when a square triangular number was found, the square
root of such is a balancing number.
Cobalancing numbers are closely related to balancing
numbers. A cobalancing number is a positive integer k
such that
1 2 ... ( 1) ( 2) ... ( )k k k kr+++= ++ + ++ +
for a
positive integer r called the cobalancer27. Some properties
of square triangular, balancing and cobalancing numbers
had been determined28,29. Congruences for prime sub-
scripted balancing numbers were established as well30.
e Pell and associated Pell numbers were also linked
with balancing and cobalancing numbers31. e concept
of balancing numbers was also employed32 to solve a
generalized Pell’s equation
2 22 2
54
y ax a
−=
.
Balancing and cobalancing numbers were also
generalized into arbitrary sequences thereby dening the
sequence balancing numbers and sequence cobalancing
numbers33. ese were further generalized into t-balanc-
ing numbers34 and sequence t-balancing numbers35. New
identities were also established together with (a,b)-type
balancing and cobalancing numbers36. Further, an analog
of balancing number, the multiplying balancing number,
was also dened37.
2.4 Triangular Number, Factorial and
Factorial-like Numbers
Probably, the most well-known analogs in number theory
are the factorials and the triangular numbers. While a tri-
angular number is the sum of positive integers, a factorial
is their product. e factorial of n, denoted by n!, gives
the number of ways in which n objects can be permuted12.
It is also the total number of essentially dierent arrange-
ments using all given n objects of distinct sizes such that
each object is suciently large to simultaneously contain
all previous objects38.
ere are also studies regarding the relationship
between factorials and triangular numbers. For instance,
there is a natural number m such that
!2
m
nT=
where
T is a product of triangular numbers and the number of
factors depends on the parity of n11.
While there are numbers that are both square and
triangular, there are also numbers that are both triangu-
lar and factorial. ese can be determined by solving the
Diophantine
( 1)
!
2
kk
n
+
=.
Some solutions, (n, k), of which are (1, 1), (3, 3) and
(5, 15). Hence, 1, 6 and 120 are numbers that are both trian-
gular and factorial. Tomaszewski in sequence A000142 of
OEIS conjectured that these are the only such numbers38.
Factorial sums, certain type of numbers expressed
as sums of factorials, and factorials expressed as sums of
other types of numbers have been studied also. e sum-
of-factorial function39 is dened by
1
() !
n
k
Sn k
=
=
∑
, which
gives the sequence }1, 3, 9, 33, 153, 873, 5913, 46233,
409113, …} or sequence A007489 in OEIS38.
Romer C. Castillo
Indian Journal of Science and Technology 5
Vol 9 (41) | November 2016 | www.indjst.org
as factoriangular numbers, and some expositions on these
numbers. While the name polygorial is a contraction of
polygonal and factorial14; factoriangular is a contraction
of factorial and triangular. As polygorial does not mean a
number that is both polygonal and factorial, a factorian-
gular number does not also mean a number that is both
factorial and triangular.
e natural similarity of factorials and triangular
numbers motivated the author to add the corresponding
numbers of the two sequences in order to form another
sequence. Adding the corresponding factorials and trian-
gular numbers gives the sequence of numbers {2, 5, 12,
34, 135, 741, 5068, 40356, 362925, …}.
Curious enough, the author checked if such sequence
is already included in the OEIS. e formulation for this
sequence is very simple and it is not surprising to nd
the sequence in OEIS, which is sequence A10129238. But
what is quite surprising is that there is very little informa-
tion about this sequence in OEIS, in particular, and in the
literature, in general. Aside from the formula and the list
of the rst 20 numbers in the sequence, provided only are
the Maple and the Mathematica programs for generating
the sequence. ere are no references indicated and no
comments from OEIS contributors and number theory
practitioners, despite the fact that the sequence is there in
OEIS for some years already. It seems that nobody has yet
explored the properties of integers that are sums of two
of the most important, most popular, and mostly studied
numbers: the factorials and the triangular numbers.
Series of experimental mathematics done by the
researcher resulted to the characterization of facto-
riangular numbers as to its parity, compositeness,
number and sum of positive divisors and other minor
characteristics44. For a positive integer k, the kth factori-
angular number, denoted by Ftk , is given by the formula
!
kk
Ft k T=+
, where
! 1 2 3 kk= ⋅ ⋅ ⋅⋅⋅
and
1 2 3 ...
k
Tk=+++ +
.
k
Ft
is even if
1k=
or
if k is of the form 4r (for integer r ≥ 1) or 4r + 3 (for integer
r ≥ 0); but it is odd if k is of the form 4r + 1 (for integer
r ≥ 1) or 4r + 2 (for integer r ≥ 0). For k = 1, 2,
k
Ft
is
prime; but for k ≥ 3,
k
Ft
is composite and it is divisible
by k if k is odd and by k/2 if k is even. Further, for even k,
there is an integer 1
r
such that 11
/ ( 1)
kk
Ft r Ft k
+
=+ +
and this
1
r
is equal to 2
( 2) / 2k−; while for odd k, there is
an integer
2
r
such that
21
2 2 / ( 1)
kk
Ft r Ft k
+
=+ +
and
this
2
r
is equal to 2
2k−
.
In another paper of the author, an exposition on the
runsum representations of factoriangular numbers was
ere are also square numbers that are sums of distinct
factorials39 like 2
3 1! 2! 3!=++
, 2
5 1! 4!=+
, 2
11 1! 5!=+
2
12 4! 5!=+
,
2
27 1! 2! 3! 6!=+++
, 2
29 1! 5! 6 !=++
,
2
71 1! 7!=+
, 2
72 4! 5! 7!=++
, 2
213 1! 2! 3! 7! 8!=++++
,
2
215 1! 4! 5! 6! 7! 8!=+++++
, 2
603 1! 2! 3! 6! 9!=++++
,
2
635 1! 4! 8! 9!=+ ++
, and
2
1183893 1! 2! 3! 7! 8! 9! 10! 11! 12! 13! 14! 15!=+++++++++++
;
triangular numbers that are sums of distinct factorials such
as 2
1! 2!T=+
, 17 1! 2! 3! 4! 5!T=++++ , 108
3!5!6!7!T=+++
,
284
3! 4! 5! 8!T=+++
, 286
1! 6! 8!T=+ +
, and 8975
5! 9 ! 11!T=++
; and triangular numbers equal to a single factorial, for
example 1
1!T=
, 3
3!T=
and
15 5!T=
.
Factorials can also be expressed as sums of Fibonacci
numbers. All factorials that are sums of at most three
Fibonacci numbers had been determined40 and it was
shown41 also that if k is xed then there are only nitely
many positive integers n such that
12
! ! ... !
nk
Fmm m= + ++
,
where Fn is a Fibonacci number, holds for some positive
integers
1, ..., k
mm
.
Factorial-like numbers also abound the literature.
ese factorial-like numbers are products of numbers
in a sequence other than the consecutive natural num-
bers from 1 to k. Some of these are the double factorials,
the primorials, and the polygorials. e product of even
numbers,
2 4 6 (2 ) (2 )!!kk⋅ ⋅ ⋅⋅⋅ =
and the product of odd
numbers,
1 3 5 (2 1) ( 2 1)!!kk⋅ ⋅ ⋅⋅⋅ − = −
are called double
factorials42. A primorial, which is an analog of the usual
factorial for prime numbers, is a product of sequential
prime numbers43. Polygorials14 are products of sequential
polygonal or n-gonal numbers, the triangular polygorials,
square polygorials, pentagonal polygorials or pentagori-
als, hexagonal polygorials or hexagorials, and so forth.
For instance, if
n
k
℘
denotes the kth n-gorial for
3n≥
and
1k≥
, then the kth triangular polygorial is given
by
3136
kk
T℘ = ⋅ ⋅ ⋅⋅⋅
and the kth square polygorial by
42
149
k
k
℘ = ⋅ ⋅ ⋅⋅⋅
. 3
k
℘
is the same as 1
2
k
k
i
T−
=
∏
, which
gives the sequence {1, 3, 18, 180, 2700, 56700, …} or
sequence A006472 in OEIS38.
3. Findings and Discussion
Much had been studied about triangular numbers,
factorials and other numbers involving sums of triangular
numbers or sums of factorials. is section now presents an
innovative study about the sums of corresponding facto-
rials and triangular numbers, which are being introduced
A Survey on Triangular Number, Factorial and Some Associated Numbers
Indian Journal of Science and Technology
6Vol 9 (41) | November 2016 | www.indjst.org
Experimental mathematics is a key tool in
discovering new kinds of numbers. However, inge-
nuity and creativeness still play a vital role. Simple
additions or multiplications of existing numbers may
lead the investigator to create or invent a new category
of numbers.
Surveys, expositions and explorations on existing
studies may continue to be a major undertaking in num-
ber theory and in mathematics in general. Conjectures
need to be proven or otherwise and open questions need
to be answered.
5. References
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Accessed:22/11/2014.
4. O’Connor JJ, Robertson EF. Quotations by Carl Friedrich
Gauss. http://www-groups.dcs.st-and.ac.uk/history/
Quotations/Gauss.html. Date Accessed:1/9/2015.
5. Burton DM. Elementary Number eory. Boston: Allyn
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6. Nguyen HD. Mathematics by experiment: Exploring patterns
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done45. A runsum is a sum of a run of consecutive positive
integers46. In determining the runsum representations of
k
Ft
, the discussion on rumsums of length k given in
another article47 served as the groundwork. It was found
out that every k
Ft
has a runsum representation of length
k, the rst term of which is
( 1)! 1k−+
and the last term
is
( 1)! kk−+
and these rst terms and last terms form
sequences similar to A038507 and A213169 of OEIS38,
respectively. Further, for k ≥ 2, the sums of the rst and
last terms of the runsums of length k of k
Ft
form another
interesting sequence with the formula,
2 ( 1) ! 1kk− ++
,
which is equal to twice the k
Ft
divided by k. In addi-
tion, two identities were established: ( 1)! ( 1) !
k kk k
Ft T T
−+ −
=−
and ( 1)! ( 1)!
!kk k k
kT T T
−+ −
= −−
, where i
T
is the ith triangular
number.
Consequently, it was found that the sequence of
factoriangular numbers is a recurring sequence with a
rational closed-form exponential generating function48.
ese numbers follow the recurrence relations
2
1
2
( 1) 2
kk
k
Ft k Ft
+
−
=+ −
, for
1k≥
, and
2
1
21
2
kk
kk
Ft k Ft −
−−
=−
, for
2k≥
.
And the exponential generating function of the
sequence of
k
Ft
is given by the formula:
2 34
2
2 (2 5 2 )
() 2(1 )
x
x x xe
Ex x
+− + +
=− ,
11x−< <
.
Moreover, interesting expositions on factoriangular
numbers expressed as sum of two triangular numbers
and/or as sum of two squares were also conducted49. It was
found that for natural numbers k, r, and s and ith triangu-
lar number i
T
, the following were established: (1) only
two solutions,
11 3 3
( , ) ( , ), ( , )
kk
Ft T Ft T Ft T=
, satisfy the
relation
2
kk
Ft T=
; (2)
2
kr
Ft T=
if and only if
41
k
Ft +
is a square; (3)
k rk
Ft T T=+
if and only if
8 ! 1k+
is a
square; and (4)
k rs
Ft T T=+
if and only if
82
k
Ft +
is a
sum of two squares.
4. Conclusion
e dierent types of numbers were among the foremost
subjects of study in the oldest times. Nevertheless, such
study on numbers is not yet complete and most probably,
will never be completed. New and fascinating groups or
sequences of numbers may be discovered in contem-
porary times and in the future by both expert number
theorists and novice number enthusiasts.
Romer C. Castillo
Indian Journal of Science and Technology 7
Vol 9 (41) | November 2016 | www.indjst.org
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