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ISSN (Print) : 0974-6846 Indian Journal of Science and Technology, Vol 9(41), DOI: 10.17485/ijst/2016/v9i41/85182, November 2016 ISSN (Online) : 0974-5645 A Survey on Triangular , and Some Associated

Romer C. Castillo College of Accountancy, Business, Economics and International Hospitality Management, Batangas State University, Batangas City – 4200., Philippines;[email protected]

Abstract Objectives: The paper aims to present a survey of both time-honored and contemporary studies on ,­ ­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, 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. of experimental resulted to the ­characterization of factoriangular­ ­numbers as to its parity, compositeness, number and sum of positive and other minor ­characteristics. It was also found that every factoriangular number has a runsum representation of length k term of which is (k −+ 1)! 1 and the last term is (k −+ 1)! k. The of factoriangular numbers is a recurring ­sequence and it has a rational closed-form of exponential­ . These numbers were also characterized, the first as to when a factoriangular­ ­number can be expressed­ as a sum of two triangular numbers and/or as a sum of two . 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 .

Keywords: Factorial, Factorial-like number, Factoriangular number, , Triangular number

1. Introduction position in the world of both ancient and contemporary mathematics. In the mathematical field, a sense of beauty seems to be As far back as ancient Greece, mathematicians were almost the only useful drive for discovery1 and it is imag- studying number theory. The Pythagoreans were very ination and not reasoning that seems to be the moving much interested in the somewhat mythical properties power for invention in mathematics2. It is in number the- of integers. They initiated the study of perfect numbers, ory that many of the greatest mathematicians in history deficient and abundant numbers, , had tried their hand3 paving the way for mathematical polygonal numbers, and Pythagorean triples. Since then, almost every major civilization had produced number experimentations, explorations and discoveries. Gauss theorists who discovered new and fascinating properties once said that the theory of numbers is the queen of of numbers for nearly every century. 4 mathematics and mathematics is the queen of science . Until today, number theory has shown its irresistible The theory of numbers concerns the characteristics appeal to professional, as well as beginning, mathemati- of integers and rational numbers beyond the ordinary cians. One reason for this lies in the basic nature of its arithmetic computations. Because of its unquestioned problems5. Although many of the number theory prob- historical importance, this theory had occupied a central lems are extremely difficult to solve and remain to be the

*Author for correspondence A Survey on Triangular Number, Factorial and Some Associated Numbers

most elusive unsolved problems in mathematics, they can There is also an identity involving the triangular be formulated in terms that are simple enough to arouse ­number and factorial. This relationship between factorials n 11,12 (2nT )!= 2n the interest and curiosity of even those without much and triangular numbers is given by : ∏ k =1 21k − mathematical training. . Aside from this, there is a somewhat natural relation on More than in any part of mathematics, the methods­ of these factorials and triangular numbers: the triangular inquiry in number theory adhere to the scientific approach. number is regarded as the additive analog of factorial. Those working on the field must rely to a large extent The current work aims to present a survey of both upon trial and error and their own curiosity, intuition time-honored and contemporary studies on triangu- and ingenuity. Rigorous mathematical proofs are often lar number and other polygonal numbers, factorial and preceded by patient and time-consuming ­mathematical factorial-like numbers, and some other related or associ- experimentation or experimental mathematics. ated numbers. It also includes some interesting results of Experimental mathematics refers to an approach of recent studies conducted by the author regarding the sum studying mathematics where a field can be effectively of corresponding factorial and triangular number, which studied using advanced computing technology such as is named as factoriangular number. computer algebra systems6. However, computer system alone is not enough to solve problems; human intuition and 2. Survey on Triangular Number, insight still play a vital role in successfully leading the math- ematical explorer on the path of discovery. And besides, Factorial and Related Numbers number theorists in the past have done these mathematical The in general and the history ­experimentations only by hand analysis and computations. of number theory in particular are inseparable. Number Among the many works in number theory that can be theory is one of the oldest fields in mathematics and most done through experimental mathematics, exploring pat- of the greatest mathematicians contributed for its devel- terns in integer is one of the most interesting and opment3. Although it is probable that the ancient Greek frequently conducted. It is quite difficult now to count the mathematicians were largely indebted to the Babylonians number of studies on sequence, , and Egyptians for a core of information about the proper- the Pell and associated Pell sequences, and other well-known ties of natural numbers, the first rudiments of an actual sequences. Classical number patterns like the triangular theory are generally credited to Pythagoras and his and other polygonal and figurate numbers have also been ­followers, the Pythagoreans5. studied from the ancient up to the modern times by mathe- maticians, professionals and amateurs alike, in almost every 2.1 Triangular Number and Other part of the world. The multiplicative analog of the triangular number, the factorial, has also a special place in the litera- Polygonal Numbers ture being very useful not only in number theory but also An important of natural numbers in ancient Greece in other mathematical disciplines like combinatorial and is the set of polygonal numbers. The name polygonal mathematical analysis. Quite recently, sequences of integers number was introduced by Heysicles to refer to positive generated by summing the digits7 were also being studied. integers that are triangular, oblong, , and the like13. Integer patterns or sequences can be described by These numbers can be considered as the ancient link ­algebraic formulas, recurrences and identities. As an between number theory and geometry.

instance, the triangular number (Tk ), for k ≥ 1, is deter- The characteristics of these polygonal numbers

mined by Tk =+ kk( 1) / 2 . Given Tk the next term in were studied by Pythagoras and the Pythagoreans. They the triangular number sequence is also determined by depicted these numbers as regular arrangements of dots

­recurrence: Tkk+1 = Tk ++1. in geometric patterns. The triangular numbers were rep- Some identities on triangular numbers are also resented as triangular array of dots, the oblong numbers 8-10 2 given in the literature, for instance : TTkk+=−1 k ; as rectangular array of dots, and the square numbers as TT+ =+( k 1)2 ; TT22− =+( k 1) 3 ; kT=+( k 2) T ; square array of dots. They also found that an oblong num- kk+1 kk+1 kk+1 22 2 ber is a sum of even numbers; a triangular number is a T2 =+ TT ; T21k=+3 TT kk− ; T2kk−=2 Tk ; and k kk−1 sum of positive integers; and a is a sum 2 13 8Tkk += 1 (2 + 1) . of odd numbers . In modern notation, if the triangular

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number is denoted by Tk , the oblong number by Ok , and ­ranging from simple relationships between them and the the square number by Sk , then other polygonal numbers to very complex relationships k involving partitions, modular forms and combinatorial 8 Tik =∑ =+1 2 ++ 3 ... + k ; properties . Many other important results on these num- i=1 bers have been discussed in the literature. A theorem of k Fermat states that a positive integer can be expressed as a Oik =∑ 2 =+++ 2 4 6 ... + 2 k; and i=1 sum of at most three triangular, four square, five pentago-

k nal, or n n-gonal numbers. Gauss proved the triangular . 10,15 Sik =∑(2 − 1) =+++ 1 3 5 ... + (2k − 1) case . Euler left important results regarding this theo- i=1 rem, which were utilized by Lagrange to prove the case 16 It is very evident that 2TOkk= . Relating ­triangular for squares and Jacobi also proved this independently . and square numbers, Nicomachus found that Cauchy showed the full proof of Fermat’s theorem15,16. 2 2 TTkk+++11 = S k =+( k 1) or Tk−1 += TSk kk =. He also The expression of an integer as a sum of three noted that triangular numbers can be produced from ­triangular numbers can be done in more than one way 13 square and oblong numbers in particular, SOTkk+=2 k and Dirichlet showed how to derive such number of 13 10 and OSkk+=++1 T 21 k. In addition, Plutarch also found ways . The modular form theory can also be used to cal- 2 that 8TSkk+= 121+ = (2 k + 1) . culate the representations of integers as sums of triangular Further, Nicomachus proved13 that, if the pentagonal numbers17. The ways a positive integer can be expressed as 5 5 number is denoted by P k , then for k >1, Pkk=+3 Tk−1 a sum of two n-sided regular figurate numbers can also be 5 18 and Pk=+ ST kk−1 . generalized . Furthermore, it was shown that a generating n If the polygonal number is be denoted by P k , where function manipulation and a combinatorial argument can n is the number of sides, then the triangular number be used on the partitions of an integer into three triangu- 3 4 can be denoted by P k and the square number by P k . lar numbers and into three distinct triangular numbers, Using these new notations, the relations among pentag- respectively19. onal, square and triangular numbers is now given, for The theory of theta functions can be used also to 5 43 k >1, by PPPk=+ kk−1 . Nicomachus generalized this compute the number of ways a can be nn−13 and claimed that, for k >1and n ≥ 3 , PPk=+ kk P−1 expressed as a sum of three squares or of three triangular holds for every n-gonal number14. This shows that every numbers20. Following that, several studies on the mixed polygonal number can be generated using the triangular sums of triangular and square numbers were conducted. numbers. Any positive integer was shown to be a sum of two For instance, squares can be generated using triangular numbers and an even square and that each nat- 22 Sk=+ TT kk−1 , as Nicomachus observed; pentagonal num- ural number is equal to a triangular number plus xy+ 5 bers using Pk=+ ST kk−−11 =+ T k2 T k; hexagonal numbers with x not congruent to y modulo 2 and where x and y are 65 21 using Pk= PT kk +−−11 =+ TT k3 k; and in general, n-go- integers . Three conjectures on mixed sums of triangular nn−1 21 nal numbers using Pk= P kk + T−−11 =+− TnT k( 3) k, and square numbers were verified , the first for n ≤15000 which is used to establish the closed formula for any kth while the second and third for n ≤10000. The second ­polygonal number14, for k ≥1 and n ≥ 3 , conjecture was later proven by using Gauss-Legendre Theorem and Jacobi’s identity22. The first conjecture had n k( kn− 2 k −+ n 4) . 23 P k = been later proven as well while the generalized Riemann 2 hypothesis implied the third conjecture about which Other established relations between triangular and explicit natural numbers may be represented15. 10 PT6 = other polygonal numbers are as follows : kk21− and Another conjecture states that for non-negative ­integers 5 1 PTkk= 3 31− . m and n, every sufficiently large positive integer can be mn22 expressed as either of the following: (1) 22x++ yTz , (2) 2.2 Contemporary Studies on Triangular mn2 mn 22n 22x++ TTyz, (3) 22Tx++ TT yz, (4) x+⋅23 yT +z , 2 n n 2 n Numbers (5) x+⋅23 TTyz +, (6) 23⋅++x 2 TTyz, (7) 23⋅+Tx 2 TT yz +, n 22 Triangular numbers, though having a simple definition,­ (8) 25⋅TTTxyz ++, (9) 234TTTxyz++, (10) 232xyT++z. are amazingly rich in properties of various kinds, Six of the ten forms: (1), (4), (5), (6), (9) and (10), were

Vol 9 (41) | November 2016 | www.indjst.org Indian Journal of Science and Technology 3 A Survey on Triangular Number, Factorial and Some Associated Numbers

proven while for the four other forms: (2), (3), (7) and (8), The Pell and associated Pell numbers were also linked counterexamples were found15. with balancing and cobalancing numbers31. The concept of balancing numbers was also employed32 to solve a 2 22 2 2.3 and Related ­generalized Pell’s equation y−=54 ax a. Numbers Balancing and cobalancing numbers were also ­generalized into arbitrary sequences thereby defining the Square triangular numbers or numbers that are both sequence balancing numbers and sequence cobalancing square and triangular are also well-studied. A square tri- numbers33. These were further generalized into t-balanc- angular number can be written as n2 for some n and as ing numbers34 and sequence t-balancing numbers35. New k(k+1)/2 for some k, and hence, is given by the equation identities were also established together with (a,b)-type kk(+ 1) balancing and cobalancing numbers36. Further, an analog n2 = 2 of balancing number, the multiplying balancing number, which is a for which integer was also defined37. solutions are demanded. Solving this equation by algebraic manipulations leads to (2kn+− 1)22 2(2 ) = 1 and by change of 2.4 Triangular Number, Factorial and 22 variables ak=+21 and bn= 2 becomes ab−=21, which Factorial-like Numbers is factorable into (a+ 2)( ba −= 2) b 1 in finding integer solu- tions. In a more general equation a22−= kb 1 where k is a Probably, the most well-known analogs in number theory fixed integer, if (,)ab is a solution then kb= ( / 2)2 is a square are the factorials and the triangular numbers. While a tri- triangular number24. angular number is the sum of positive integers, a factorial The sequence of square triangular numbers is is their product. The factorial of n, denoted by n!, gives 12 {1, 36, 1225, 41616, 1413721, 48024900, 1631432881, the number of ways in which n objects can be permuted . 55420693056, …}. Thek th square triangular number, It is also the total number of essentially different arrange-

which can be denoted by Stk , can be obtained from the ments using all given n objects of distinct sizes such that

recursive formula, Stk=34 St kk−−12 −+ St 2 for k ≥ 3 with St1 =1 each object is sufficiently large to simultaneously contain 38 and St2 = 36 . The non-recursive formula, for k ≥1, all previous objects . There are also studies regarding the relationship 22kk2 12+ −− 12 between factorials and triangular numbers. For instance, ( ) ( ) m Stk = = 42 there is a natural number m such that nT!2 where  T is a product of triangular numbers and the number of also gives the kth square triangular number25. factors depends on the parity of n11. Square triangular numbers are related to While there are numbers that are both square and ­balancing ­numbers. A balancing number is triangular, there are also numbers that are both triangu- a positive integer k that makes the equation lar and factorial. These can be determined by solving the 1+++−=++++++ 2 ... (k 1) ( k 1) ( k 2) ... ( kr ) true for a Diophantine positive integer r called balancer26. The equation is equiv- kk(+ 1) n!= . alent to TTTk−+1 =− kr k or Tk−+1 += TT k kr, which is also the 2 2 same as k=( krkr + )(( ++ 1) / 2 . It is clear that when k2 is a triangular number, k is a balancing number. Hence, Some solutions, (n, k), of which are (1, 1), (3, 3) and when a square triangular number was found, the square (5, 15). Hence, 1, 6 and 120 are numbers that are both trian- root of such is a balancing number. gular and factorial. Tomaszewski in sequence A000142 of Cobalancing numbers are closely related to balancing OEIS conjectured that these are the only such numbers38. numbers. A cobalancing number is a positive integer k Factorial sums, certain type of numbers expressed such that 1+++= 2 ...k ( k ++ 1) ( k + 2) ++ ... ( kr + ) for a as sums of factorials, and factorials expressed as sums of positive integer r called the cobalancer27. Some properties other types of numbers have been studied also. The sum- n of square triangular, balancing and cobalancing numbers 39 Sn()= k ! of-factorial function is defined by ∑ k =1 , which had been determined28,29. Congruences for prime sub- gives the sequence }1, 3, 9, 33, 153, 873, 5913, 46233, scripted balancing numbers were established as well30. 409113, …} or sequence A007489 in OEIS38.

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There are also square numbers that are sums of distinct as factoriangular numbers, and some expositions­ on these 2 factorials39 like 32 =+ 1! 2! + 3! , 52 =+ 1! 4!, 11=+ 1! 5! numbers. While the name polygorial is a contraction of 14 2 2 2 polygonal and factorial ; factoriangular is a contraction 12=+ 4! 5!, 27=+ 1! 2! ++ 3! 6!, 29=++ 1! 5! 6!, 2 2 2 of factorial and triangular. As polygorial does not mean a 71=+ 1! 7! 72=++ 4! 5! 7! 213=++++ 1! 2! 3! 7! 8! , , , number that is both polygonal and factorial, a factorian- 2152 =+++++ 1! 4! 5! 6! 7! 8! 6032 =+ 1! 2! +++ 3! 6! 9! , , gular number does not also mean a number that is both 2 2 635=+ 1! 4! ++ 8! 9! , and 1183893=+++++++++++ 1! 2! 3! 7! 8! 9!; 10! 11!factorial 12!13! and 14!triangular. 15! triangular numbers that are sums of distinct factorials such The natural similarity of factorials and triangular as T2 =+1! 2!, T17 =++++1! 2! 3! 4! 5! , T108 =+++3!5!6!7!, numbers motivated the author to add the corresponding T284 =+++3! 4! 5! 8!, T286 =++1! 6! 8!, and T8975 =++5! 9! 11! numbers of the two sequences in order to form another ; and triangular numbers equal to a single factorial, for sequence. Adding the corresponding factorials and trian- T = 5! example T1 =1!, T3 = 3! and 15 . gular numbers gives the sequence of numbers {2, 5, 12, Factorials can also be expressed as sums of Fibonacci 34, 135, 741, 5068, 40356, 362925, …}. numbers. All factorials that are sums of at most three Curious enough, the author checked if such sequence Fibonacci numbers had been determined40 and it was is already included in the OEIS. The formulation for this shown41 also that if k is fixed then there are only finitely sequence is very simple and it is not surprising to find many positive integers n such that the sequence in OEIS, which is sequence A10129238. But what is quite surprising is that there is very little informa- Fmmnk=12! + ! ++ ... m !, tion about this sequence in OEIS, in particular, and in the where F is a , holds for some positive n literature, in general. Aside from the formula and the list mm, ..., integers 1 k . of the first 20 numbers in the sequence, provided only are Factorial-like numbers also abound the literature. the Maple and the Mathematica programs for generating These factorial-like numbers are products of numbers the sequence. There are no references indicated and no in a sequence other than the consecutive natural num- comments from OEIS contributors and number theory bers from 1 to k. Some of these are the double factorials, practitioners, despite the fact that the sequence is there in the , and the polygorials. The product of even OEIS for some years already. It seems that nobody has yet 2⋅ 4 ⋅ 6 ⋅⋅⋅ (2kk ) = (2 )!! numbers, and the product of odd explored the properties of integers that are sums of two 1⋅ 3 ⋅ 5 ⋅⋅⋅ (2kk − 1) = (2 − 1)!! numbers, are called double of the most important, most popular, and mostly studied 42 factorials . A , which is an analog of the usual numbers: the factorials and the triangular numbers. factorial for prime numbers, is a product of sequential Series of experimental mathematics done by the 43 14 prime numbers . Polygorials are products of sequential researcher resulted to the characterization of facto- polygonal or n-gonal numbers, the triangular polygorials, riangular numbers as to its parity, compositeness, square polygorials, pentagonal polygorials or pentagori- number and sum of positive divisors and other minor als, hexagonal polygorials or hexagorials, and so forth. 44 th n characteristics . For a positive integer k, the k factori- ℘ th For instance, if k denotes the k n-gorial for n ≥ 3 angular number, denoted by Ft , is given by the formula and k ≥1, then the kth triangular polygorial is given k Ftkk =+ k! T , where kk! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ and by ℘3 =136 ⋅ ⋅ ⋅⋅⋅T and the kth square polygorial by kk Tkk =+ 1 2 ++ 3 ... + . Ft is even if k = 1 or k k 423 T ℘k =149 ⋅ ⋅ ⋅⋅⋅k . ℘ k is the same as ∏i=2 k −1 , which if k is of the form 4r (for integer r ≥ 1) or 4r + 3 (for integer gives the sequence {1, 3, 18, 180, 2700, 56700, …} or r ≥ 0); but it is odd if k is of the form 4r + 1 (for integer 38 sequence A006472 in OEIS . r ≥ 1) or 4r + 2 (for integer r ≥ 0). For k = 1, 2, Ftk is prime; but for k ≥ 3, Ftk is composite and it is divisible 3. Findings and Discussion by k if k is odd and by k/2 if k is even. Further, for even k, there is an integer r1 such that Ftkk=+ r11 Ft+ / ( k + 1) 2 Much had been studied about triangular numbers, and this r1 is equal to (k − 2) / 2 ; while for odd k, there is

­factorials and other numbers involving sums of triangular an integer r2 such that 2Ftkk=+ r21 2 Ft+ / ( k + 1) and 2 numbers or sums of factorials. This section now presents an this r2 is equal to k − 2 . innovative study about the sums of corresponding facto- In another paper of the author, an exposition on the rials and triangular numbers, which are being ­introduced runsum representations of factoriangular numbers was

Vol 9 (41) | November 2016 | www.indjst.org Indian Journal of Science and Technology 5 A Survey on Triangular Number, Factorial and Some Associated Numbers

done45. A runsum is a sum of a run of consecutive positive Experimental mathematics is a key tool in integers46. In determining the runsum representations of ­discovering new kinds of numbers. However, inge-

Ftk , the discussion on rumsums of length k given in nuity and creativeness still play a vital role. Simple another article47 served as the groundwork. It was found additions or multiplications of existing numbers may

out that every Ftk has a runsum representation of length lead the investigator­ to create or invent a new category k, the first term of which is (k −+ 1)! 1 and the last term of ­numbers. is (kk−+ 1)! and these first terms and last terms form Surveys, expositions and explorations on existing sequences similar to A038507 and A213169 of OEIS38, studies may continue to be a major undertaking in num- respectively. Further, for k ≥ 2, the sums of the first and ber theory and in mathematics in general. Conjectures

last terms of the runsums of length k of Ftk form another need to be proven or otherwise and open questions need interesting sequence with the formula, 2(kk− 1)! ++ 1 , to be answered.

which is equal to twice the Ftk divided by k. In addi-

tion, two identities were established: Ftk=− T( kk−+ 1)! T ( k − 1)! 5. References and kT!= −− T T, where T is the ith triangular (kk−+ 1)! ( k − 1)! k i 1. Hadamard J. The Psychology of Invention in the number. Mathematical Field. New York: Dover Publications; 1954. Consequently, it was found that the sequence of 2. De Morgan A. Sir W. R. Hamilton. Gentleman’s Magazine ­factoriangular numbers is a recurring sequence with a and Historical Review. 1866; 1: 128–34. rational closed-form exponential generating function48. 3. Heck B. Number theory. http://math.nicholls.edu/heck/ These numbers follow the recurrence relations teaching/573/573(Hom)Su09/NumberTheory.pdf. Date 2 k − 2 , for k ≥1, and Accessed:22/11/2014. Ftkk+1 =+( k 1)  Ft − 2 4. O’Connor JJ, Robertson EF. Quotations by . http://www-groups.dcs.st-and.ac.uk/history/ kk2 −−21 Ftkk=− k Ft −1 , for k ≥ 2 . Quotations/Gauss.html. Date Accessed:1/9/2015. 2  5. Burton DM. Elementary Number Theory. Boston: Allyn And the exponential generating function of the and Bacon; 1980.

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