Total views : 655

A Survey on Triangular Number, Factorial and Some Associated Numbers


  • College of Accountancy, Business, Economics and International Hospitality Management, Batangas State University, Batangas City – 4200, Philippines


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 (k −1)! + 1 and the last term is (k −1)! + k . 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.


Factorial, Factorial-like number, Factoriangular number, Polygonal number, Triangular number.

Full Text:

 |  (PDF views: 367)


  • Hadamard J. The Psychology of Invention in the Mathematical Field. New York: Dover Publications; 1954.
  • De Morgan A. Sir W. R. Hamilton. Gentleman’s Magazine and Historical Review. 1866; 1: 128–34.
  • Heck B. Number theory. Date Accessed:22/11/2014.
  • O’Connor JJ, Robertson EF. Quotations by Carl Friedrich Gauss. Date Accessed:1/9/2015.
  • Burton DM. Elementary Number Theory. Boston: Allyn and Bacon; 1980.
  • Nguyen HD. Mathematics by experiment: Exploring patterns of integer sequences., Date Accessed:22/11/2014.
  • Okagbue HI, Adamu MO, Iyase SA, Opanuga AA. Sequence of integers generated by summing the digits of their squares. Indian Journal of Science and Technology. 2015 Jul; 8(15):1–7.
  • Garge AS, Shirali SA. Triangular numbers. Resonance. 2012 Jul;, 17(7): 672-81.
  • Hoggatt VE, Bicknell M. Triangular numbers. Fibonacci Quarterly. 1974 Oct; 12: 221-30.
  • Weisstein EW. Triangular number. Date Accessed:21/12/2014.
  • Florez R, Junes L. A relation between triangular numbers and prime numbers. Integers., Date Accessed:20/1/2015;11:1–12.
  • Weisstein EW. Factorial., Date Accessed:21/12/2014.
  • Tattersall JJ. Elementary Number Theory in Nine Chapters. Cambridge: Cambridge University Press; 1999.
  • Dockery D. Polygorials: Special factorials of polygonal numbers., Date Accessed:11/11/2014.
  • Kane B. On two conjectures about mixed sums of squares and triangular numbers. Journal of Combinatorics and Number Theory. 2009; 1(1):75–88.
  • Leroux P. A simple symmetry generating operads related to rooted planar m-nary trees and polygonal numbers. Journal of Integer Sequences., Date Accessed:12/12/2014.
  • Ono K, Robins S, Wahl PT. On the representation of integers as sums of triangular numbers. Aequationes Mathematicae. 1995; 50(1-2): 3–94.
  • Rothschild, SJ. The number of ways to write n as a sum of ℓ regular figurate numbers [Undergraduate honors capstone project]. New York: Syracuse University., Date Accessed:22/11/2014.
  • Hirschhorn MD, Sellers JA. Partitions into three triangular numbers. Australasian Journal of Combinatorics. 2004; 30:307–18.
  • Farkas HM. Sums of squares and triangular numbers. Online Journal of Analytic Combinatorics., Date Accessed:5/5/2015.
  • Sun ZW. Mixed sums of squares and triangular numbers. Acta Arithmetica. 2007; 127(2): 103–13.
  • Guo S, Pan H, Sun ZW. Mixed sums of squares and triangular numbers (II), Integers: Electronic Journal of Combinatorics and Number Theory., Date Accessed:20/1/2015.
  • Oh BK, Sun ZW. Mixed sums of squares and triangular numbers (III). Journal of Number Theory. 2009; 129(4):964–9.
  • Carlson J. Square-triangular numbers. Date Accessed:12/11/2004.
  • Gupta SS. Fascinating triangular numbers. Date Accessed:20/1/2015.
  • Behera A, Panda GK. On the square roots of triangular numbers. Fibonacci Quarterly. 1999 May; 37: 98-105.
  • Ray PK. Balancing and cobalancing numbers [PhD thesis], Roukela: National Institute of Technology. Date Accessed:22/11/2014.
  • Olajos P. Properties of balancing, cobalancing and generalized balancing numbers. Annales Mathematicae et Informaticae.2010; 37:125–38.
  • Keskin R, Karaatli O. Some new properties of balancing numbers and square triangular numbers. Journal of Integer Sequences. Date Accessed:12/12/2015.
  • Ray PK. Curious congruences for balancing numbers. International Journal of Contemporary Mathematical Sciences. 2012; 7(18):881–9.
  • Panda GK, Ray PK. Some links of balancing and cobalancing numbers with Pell and associated Pell numbers. Bulletin of the Institute of Mathematics Academia Sinica. 2011; 6(1):41–72.
  • Dash KK, Ota RS, Dash S. Application of balancing numbers in effectively solving generalized Pell’s equation. International Journal of Scientific and Innovative Mathematical Research. 2014; 2(2):156–64.
  • Panda GK. Sequence balancing and cobalancing numbers. Fibonacci Quarterly. 2007 Oct; 45:265–71.
  • Dash KK, Ota RS, Dash S. t-Balancing numbers. International Journal of Contemporary Mathematical Sciences. 2012; 7(41):1999–2012.
  • Dash KK, Ota RS, Dash S. Sequence t-balancing numbers.International Journal of Contemporary Mathematical Sciences. 2012;7(47):2305–10.
  • Sadek B, Ali D. Some results on balancing, cobalancing, (a,b)-type balancing and (a,b)-type cobalancing numbers, Integers. Date Accessed:20/1/2013.
  • Szakacs T. Multiplying balancing numbers, Acta Univ. Sapientiae. Mathematicae., 2011; 3(1):90–6.
  • Sloane NJA. The On-line encyclopedia of integer sequences. Date Accessed:15/5/2015.
  • Weisstein EW. Factorial sums. Date Accessed:21/12/2014.
  • Luca F, Siksek S. On factorials expressible as sums of at most three Fibonacci numbers. Proceedings of the Edinburgh Mathematical Society Series 2. 2010; 53(3):747–63.
  • Grossman G, Luca F. Sums of factorials in binary recurrence sequences., Journal of Number Theory. 2002; 93:87–107.
  • Sloane NJA. A Handbook of Integer Sequences. London: Academic Press; 1973. 1–81.
  • Weisstein EW. Primorial. Date Accessed:21/12/2014.
  • Castillo RC. On the sum of corresponding factorials and triangular numbers: Some preliminary results. Asia Pacific Journal of Multidisciplinary Research. 2015; 3(4-1):5–11.
  • Castillo RC. On the sum of corresponding factorials and triangular numbers: Runsums, trapezoids and politeness. Asia Pacific Journal of Multidisciplinary Research. 2015;3(4-2): 95–101.
  • Knott R. Introducing runsums., Date Accessed:5/7/2013, 15/12/2014.
  • Knott R. More about runsums. Date Accessed:23/11/2009,16/12/2014.
  • Castillo RC. Recurrence relations and generating functions of the sequence of sums of corresponding factorials and triangular numbers.Asia Pacific Journal of Multidisciplinary Research. 2015; 3(4-3):165–69.
  • Castillo RC. Sums of two triangulars and of two squares associated with sum of corresponding factorial and corresponding number.Asia Pacific Journal of Multidisciplinary Research. 2015; 3(4-3):28–36.


  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.