Edelweiss Applied Science and Technology 
ISSN: 2576-8484 
Vol. 9, No. 3, 891-904 
2025 
Publisher: Learning Gate 
DOI: 10.55214/25768484.v9i3.5378 
© 2025 by the author; licensee Learning Gate 

© 2025 by the author; licensee Learning Gate 
History: Received: 1 January 2025; Revised: 17 February 2025; Accepted: 17 February 2025; Published: 12 March 2025 
* Correspondence:  romer.castillo@g.batstate-u.edu.ph 

 
 
 
 
 

Mathematical explorations on the sequence of factoriangular numbers: 
Extending the results on generalizations 

 
Romer C. Castillo1* 

1Batangas State University, Batangas City, Philippines; romer.castillo@g.batstate-u.edu.ph (R.C.C.). 

 

 

Abstract: A factoriangular number is formed by adding a factorial and a triangular number. If 
corresponding factorials and triangular numbers are added, the results are n-factoriangular numbers. 
Other factoriangular numbers are called (n,k)-factoriangular numbers, n(m)-factoriangular numbers, 
(n(m),k(m))-factoriangular numbers, and (n(a),k(b))-factoriangular numbers. The main objective of this study is 
to explore the sequence of (n(m),k(m))-factoriangular numbers and the sequence of (n(a),k(b))-factoriangular 
numbers as generalizations of the sequence of n-factoriangular numbers. This research is a discipline-
based scholarship of discovery that employs an exploratory method involving the scientific approach of 
experimental mathematics. The mathematical method was used in doing the explorations, focusing on 
the formulations and proofs of theorems and giving some examples. For the main results, ten theorems 
were proven and several examples of sequences were provided. The theorems include several formulas 
for (n(m),k(m))-factoriangular numbers, and (n(a),k(b))-factoriangular numbers. The proofs for theorems in 
(n(m),k(m))-factoriangular numbers are applicable for similar theorems in (n(a),k(b))-factoriangular numbers. 
Specific sequences of some generalized factoriangular numbers were presented in tables. Entries of 
numbers in the tables may lead to the formation of triangular arrays of factoriangular numbers that may 
be further explored by other researchers, especially those mostly interested in recreational mathematics. 

Keywords: Factorial, Factoriangular number, Generalization, Integer sequence, Number theory, Triangular number. 

 
1. Introduction  

Mathematics literature provide a long history of research in triangular numbers and factorials. It is 
commonly argued that triangular numbers were already known to the ancient Greeks, who viewed them 
with reverence [1] and most especially to the Pythagoreans who discoursed on the number of dots or 
pebbles that could form geometrical figures, such as a triangle [2]. While the triangular numbers were 
known to the Pythagoreans of ancient Greece, the factorials were known to the Jains of ancient India 
and to the Hebrews of ancient Middle East. Although Greek mathematics included combinatorics, there 
is no direct evidence of ancient Greek study of factorials. It is in about mid-17th to the early 18th 
century that the factorial function was intensively studied by leading mathematicians of the period 
including [3-5]. The literature also provides some expositions on triangular numbers [1] early works 
on factorial function [3] and some early and recent studies on triangular numbers, factorials, and other 
related numbers [6]. 

Generalization is an important part of mathematics and it serves as a tool in constructing new 
knowledge [7]. Mathematical generalization encompasses a claim that some property or techniques 
holds for a set of mathematical objects or conditions, the scope of which is always larger than the set of 
individually verified cases [8]. Like any other theorem, a generalization is accepted to be true if and 
only if it is supported by a valid proof. Generalizations have been applied to a variety of number-
theoretic problems. The triangular numbers, the factorials, and many number-theoretic theorems 

https://orcid.org/0000-0002-5797-7854
mailto:romer.castillo@g.batstate-u.edu.ph


892 

 

 

Edelweiss Applied Science and Technology 
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Vol. 9, No. 3: 891-904, 2025 
DOI: 10.55214/25768484.v9i3.5378 
© 2025 by the author; licensee Learning Gate 

 

related to them have been generalized in several ways in the previous studies. For instance, an 
expository article discusses some generalized factorials [9]. A more recent article generalizes triangular 
numbers to arbitrary higher-dimensional spaces [10].  

Triangular numbers and factorials are associated in such a way that triangular numbers are the 
additive analogs of the factorials [11]. The apparent natural connection between these two sequences of 
numbers contributed to the interest of adding corresponding factorials and triangular numbers to form 
a new sequence of integers, which is called factoriangular numbers [12]. This relatively new sequence is 
included in The Online Encyclopedia of Integer Sequences (OEIS) as Entry A101292 [13]. With the 
introduction of factoriangular numbers, the literature now provides Fibonacci factoriangular numbers 
[14] Pell factoriangular numbers [15] Lucas factoriangular numbers [16] factoriangular numbers in 
balancing and Lucas-balancing sequence [17] and multiple factoriangular numbers [18]. 

The multiple factoriangular number is a generalization of the factoriangular number. Several 
articles also discuss some other generalizations of factoriangular numbers [19, 20]. In this expository 
paper, we provide some explorations on further generalizations of factoriangular numbers.  
 

2. Methodology 
This expository article is a result of discipline-based scholarship of discovery particularly, basic 

research in number theory. We employ an exploratory method involving the scientific approach of 
experimental mathematics. Experimental mathematics is the methodology of doing mathematics that 
includes the use of computations for gaining insight and intuition,  discovering new patterns and 
relationships,  using graphical displays to suggest underlying mathematical principles,  testing and 
especially falsifying conjectures,  exploring a possible result to see if it is worth formal proof, suggesting 
approaches for formal proof,  replacing lengthy hand derivations with computer-based derivations, and  
confirming analytically derived results [21, 22]. More particularly, we use the mathematical method 
[22] presented below: 
 

 
Figure 1.  
Mathematical method. 

 



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Edelweiss Applied Science and Technology 
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Vol. 9, No. 3: 891-904, 2025 
DOI: 10.55214/25768484.v9i3.5378 
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In the next section, we present some results of the previous studies as preliminaries. We then 
extend the results of previous studies on the generalizations of factoriangular numbers to provide some 
explorations on the further generalizations of factoriangular numbers as the main results.  

 

3. Results and Discussion 
3.1. Preliminaries 

A number that is a sum of a factorial and its corresponding triangular number is referred to as 
factoriangular number [10]. The factoriangular number is formally defined as follows:  
Definition 3.1: The nth factoriangular number is defined by the formula 

𝐹𝑡𝑛 = 𝑛!  + 𝑇𝑛 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛 and 𝑇𝑛 = 1 + 2 + 3+. . . +𝑛 = 𝑛(𝑛 + 1)/2. 

The first few factoriangular numbers are given in the sequence {2,
5, 12, 34, 135, 741, 5068, 40356, 362925, 3628855, … }. This sequence, with 2 (i.e., 1! + T1) as the first 
term, appeared in OEIS as A101292 in 2004. In 2016, 1 (i.e., 0! + T0) was appended as the first term 
[13]. We call the numbers in this sequence n-factoriangular numbers.  

The terms of the sequence of factoriangular numbers {𝐹𝑡𝑛} for natural numbers 𝑛 ≥ 1 are of the 
form 

𝐹𝑡𝑛 = (1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛) + (1 + 2 + 3+. . . +𝑛). 
There are several ways of generalizing this sequence. 

A generalization of the sequence of n-factoriangular numbers [19] is the sequence {𝐹𝑡𝑛,𝑘} for 

natural numbers 𝑛, 𝑘 ≥ 1, which follow the form 

𝐹𝑡𝑛,𝑘 = (1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛) + (1 + 2 + 3+. . . +𝑘) 

We call the numbers in this sequence (𝑛, 𝑘)-factoriangular numbers and define as follows: 

Definition 3.2: The (𝑛, 𝑘)-factoriangular number is defined by the formula 

 𝐹𝑡𝑛,𝑘 = 𝑛!  +  𝑇𝑘 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛 and 𝑇𝑘 = 1 + 2 + 3+. . . +𝑘 = 𝑘(𝑘 + 1)/2 for natural numbers 𝑛, 𝑘 ≥ 1. 

From Definitions 3.1 and 3.2, when 𝑛 = 𝑘, the (𝑛, 𝑘)-factoriangular numbers are the same as the n-

factoriangular numbers or 𝐹𝑡𝑛,𝑘 = 𝐹𝑡𝑛.  

The sequence of the (𝑛, 𝑘)-factoriangular numbers is given by 

{𝐹𝑡𝑛,𝑘} = {2, 3, 4, 5, 7, 7, 9, 8, 12, 25, 11, 27, 12, 30, 16, 34, … } 

for (n,k) = (1,1), (2,1), (1,2), (2,2), (3,1), (1,3), (3,2), (2,3), (3,3), (4,1), (1,4), (4,2), (2,4), (4,3), (3,4), (4,4), …  

Notice that in this sequence of (𝑛, 𝑘)-factoriangular numbers, the term in the 1st, 4th, 9th, 16th, and so 
on are the n-factoriangular numbers. These terms are the entries in the main diagonal from the top-left 
to the bottom-right (i.e, when n = k) of the following table: 

 
Table 1.  
Table of (n,k)-factoriangular numbers.  

𝒏 \ 𝒌 1 2 3 4 5 6 7  𝒌 
1 2 4 7 11 16 22 29  1 + 𝑇𝑘 
2 3 5 8 12 17 23 30  2 + 𝑇𝑘 
3 7 9 12 16 21 27 34  6 + 𝑇𝑘 
4 25 27 30 34 39 45 52  24 + 𝑇𝑘 
5 121 123 126 130 135 141 148  120 + 𝑇𝑘 
6 721 723 726 730 735 741 748  720 + 𝑇𝑘 
7 5041 5043 5046 5050 5055 5061 5068  5040 + 𝑇𝑘 

          

𝑛 𝑛! + 1 𝑛! + 3 𝑛! + 6 𝑛! + 10 𝑛! + 15 𝑛! + 21 𝑛! + 28  𝑛! + 𝑇𝑘 

 



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The n-factoriangular numbers were also generalized into multiple factoriangular numbers [18] to 
have 

𝐹𝑡(𝑛, 𝑘) = (𝑛!)𝑘  +  ∑ 𝑛𝑘 

wherein, ∑ 𝑛𝑘 = 𝑇𝑛(𝑘). This is similar to a generalization of the sequence of n-factoriangular numbers 

[20] into the sequence {𝐹𝑡𝑛(𝑚)} for natural numbers 𝑛, 𝑚 ≥ 1, which follow the form 

𝐹𝑡𝑛(𝑚) = (1𝑚 ⋅ 2𝑚 ⋅ 3𝑚 ⋅⋅⋅ 𝑛𝑚) + (1𝑚 + 2𝑚 + 3𝑚+. . . +𝑛𝑚) 

We call the numbers in this sequence as 𝑛(𝑚)-factoriangular numbers and define as follows: 

Definition 3.3: The 𝑛(𝑚)-factoriangular number is defined by the formula 

 𝐹𝑡𝑛(𝑚) = (𝑛!)𝑚 + 𝑆𝑚(𝑛) 

where (𝑛!)𝑚 = 1𝑚 ⋅ 2𝑚 ⋅ 3𝑚 ⋅⋅⋅ 𝑛𝑚 and 𝑆𝑚(𝑛) = 1𝑚 + 2𝑚 + 3𝑚+. . . +𝑛𝑚 for natural numbers , 1n m 

. 

From the Definitions 3.1 and 3.3, when 𝑚 = 1, the 𝑛(𝑚)-factoriangular numbers are the same as the 

n-factoriangular numbers or 𝐹𝑡𝑛(𝑚) = 𝐹𝑡𝑛. Here, 𝑆1(𝑛) is the same as 𝑇𝑛, and thus,  

𝐹𝑡𝑛(1) = 𝑛!  +  𝑆1(𝑛) = (1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛) + (1 + 2 + 3+. . . +𝑛) = 𝑛! +
𝑛(𝑛 + 1)

2
= 𝐹𝑡𝑛 

For the next few specific cases of 𝑛(𝑚)-factoriangular numbers, that is when 𝑚 = 2, 3, 4, 5, 6, 7, 8, 9, 
we have 

𝐹𝑡𝑛(2) = (𝑛!)2 + 𝑆2(𝑛) = (12 ⋅ 22 ⋅ 32 ⋅⋅⋅ 𝑛2) + (12 + 22 + 32+. . . +𝑛2) 

𝐹𝑡𝑛(3) = (𝑛!)3 + 𝑆3(𝑛) = (13 ⋅ 23 ⋅ 33 ⋅⋅⋅ 𝑛3) + (13 + 23 + 33+. . . +𝑛3) 

𝐹𝑡𝑛(4) = (𝑛!)4 + 𝑆4(𝑛) = (14 ⋅ 24 ⋅ 34 ⋅⋅⋅ 𝑛4) + (14 + 24 + 34+. . . +𝑛4) 

𝐹𝑡𝑛(5) = (𝑛!)5 + 𝑆5(𝑛) = (15 ⋅ 25 ⋅ 35 ⋅⋅⋅ 𝑛5) + (15 + 25 + 35+. . . +𝑛5) 

𝐹𝑡𝑛(6) = (𝑛!)6 + 𝑆6(𝑛) = (16 ⋅ 26 ⋅ 36 ⋅⋅⋅ 𝑛6) + (16 + 26 + 36+. . . +𝑛6) 

𝐹𝑡𝑛(7) = (𝑛!)7 + 𝑆7(𝑛) = (17 ⋅ 27 ⋅ 37 ⋅⋅⋅ 𝑛7) + (17 + 27 + 37+. . . +𝑛7) 

𝐹𝑡𝑛(8) = (𝑛!)8 + 𝑆8(𝑛) = (18 ⋅ 28 ⋅ 38 ⋅⋅⋅ 𝑛8) + (15 + 28 + 38+. . . +𝑛8) 

𝐹𝑡𝑛(9) = (𝑛!)9 + 𝑆9(𝑛) = (19 ⋅ 29 ⋅ 39 ⋅⋅⋅ 𝑛9) + (19 + 29 + 39+. . . +𝑛9) 

Theorem 3.4: For natural number 1n  , the  𝑛(2)-, 𝑛(3)-, 𝑛(4)-,  𝑛(5)-, 𝑛(6)-, 𝑛(7)-, 𝑛(8)-, and 𝑛(9)-
factoriangular numbers are, respectively, given by the formulas 

𝐹𝑡𝑛(2) = (𝑛!)2 +
1

3
(2𝑛 + 1)𝑇𝑛 

𝐹𝑡𝑛(3) = (𝑛!)3 + 𝑇𝑛
2 

𝐹𝑡𝑛(4) = (𝑛!)4 +
1

15
(6𝑛3 + 9𝑛2 + 𝑛 − 1)𝑇𝑛  

𝐹𝑡𝑛(5) = (𝑛!)5 +
1

3
(2𝑛2 + 2𝑛 − 1)𝑇𝑛

2 

𝐹𝑡𝑛(6) = (𝑛!)6 +
1

21
(6𝑛5 + 15𝑛4 + 6𝑛3 − 6𝑛2 − 𝑛 + 1)𝑇𝑛 

𝐹𝑡𝑛(7) = (𝑛!)7 +
1

6
(3𝑛4 + 6𝑛3 − 𝑛2 − 4𝑛 + 2)𝑇𝑛

2 

𝐹𝑡𝑛(8) = (𝑛!)8 +
1

45
(10𝑛7 + 35𝑛6 + 25𝑛5 − 25𝑛4 − 17𝑛3 + 17𝑛2 + 3𝑛 − 3)𝑇𝑛 

𝐹𝑡𝑛(9) = (𝑛!)9 +
1

5
(2𝑛6 + 6𝑛5 + 𝑛4 − 8𝑛3 + 𝑛2 + 6𝑛 − 3)𝑇𝑛

2 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛 and 𝑇𝑛 = 1 + 2 + 3+. . . +𝑛 = 𝑛(𝑛 + 1)/2. 
A recent paper [20] gives the proof of Theorem 3.4. The proof has verified the following: 

(𝑛 + 1)2 − 1 = 2𝑆1 + 𝑛 

(𝑛 + 1)3 − 1 = 3𝑆2 + 3𝑆1 + 𝑛 



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(𝑛 + 1)4 − 1 = 4𝑆3 + 6𝑆2 + 4𝑆1 + 𝑛 

(𝑛 + 1)5 − 1 = 5𝑆4 + 10𝑆3 + 10𝑆2 + 5𝑆1 + 𝑛 

(𝑛 + 1)6 − 1 = 6𝑆5 + 15𝑆4 + 20𝑆3 + 15𝑆2 + 6𝑆1 + 𝑛 

(𝑛 + 1)7 − 1 = 7𝑆6 + 21𝑆5 + 35𝑆4 + 35𝑆3 + 21𝑆2 + 7𝑆1 + 𝑛 

(𝑛 + 1)8 − 1 = 8𝑆7 + 28𝑆6 + 56𝑆5 + 70𝑆4 + 56𝑆3 + 28𝑆2 + 8𝑆1 + 𝑛 

(𝑛 + 1)9 − 1 = 9𝑆8 + 36𝑆7 + 84𝑆6 + 126𝑆5 + 126𝑆4 + 84𝑆3 + 36𝑆2 + 9𝑆1 + 𝑛 

(𝑛 + 1)10 − 1 = 10𝑆9 + 45𝑆8 + 120𝑆7 + 210𝑆6 + 252𝑆5 + 210𝑆4 + 120𝑆3 + 45𝑆2 + 10𝑆1 + 𝑛 
For ease of writing of the above and to avoid confusion as to whether a function or a multiplication, the 

sum of powers of natural numbers n has been written simply as 𝑆𝑚 instead of 𝑆𝑚(𝑛) (e.g., 𝑆1 instead of 

𝑆1(𝑛)).  

The sequences of 𝑛(𝑚)-factoriangular numbers, for some specific 𝑚 ≥ 1, are given as follows: 

{𝐹𝑡𝑛(1)} = {2, 5, 12, 34, 135, . . . } for 𝑚 = 1 

{𝐹𝑡𝑛(2)} = {2, 9, 50, 606, 14455, . . . } for 𝑚 = 2 

{𝐹𝑡𝑛(3)} = {2, 17, 252, 13924, 1728225, . . . } for 𝑚 = 3 

{𝐹𝑡𝑛(4)} = {2, 33, 1394, 332130, 207360979, . . . } for 𝑚 = 4 

{𝐹𝑡𝑛(5)} = {2, 65, 8052, 7963924, 24883204425, . . . } for 𝑚 = 5 

{𝐹𝑡𝑛(6)} = {2, 129, 47450, 191107866, 2985984020515, . . . } for 𝑚 = 6 

{𝐹𝑡𝑛(7)} = {2, 257, 282252, 4586490124, 358318080096825, . . . } for 𝑚 = 7 

{𝐹𝑡𝑛(8)} = {2, 513, 1686434, 11062023330, 1334357900462979, . . . } for 𝑚 = 8 

{𝐹𝑡𝑛(9)} = {2, 1025, 10097892, 286607822564, 559780352002235465, . . . } for 𝑚 = 9 

We present the next theorem for the general case of the 𝑛(𝑚)-factoriangular numbers. 

Theorem 3.5: For natural numbers 𝑛, 𝑚 ≥ 1, the 𝑛(𝑚)-factoriangular numbers can be determined by the 
formula  

𝐹𝑡𝑛(𝑚) = (𝑛!)𝑚 +
1

𝑚 + 1
[(𝑛 + 1)[(𝑛 + 1)𝑚 − 1] − ∑ (

𝑚 + 1
𝑖

)

𝑚−1

𝑖=1

𝑆𝑖(𝑛)] 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛 and 𝑆𝑖(𝑛) = 1𝑖 + 2𝑖 + 3𝑖+. . . +𝑛𝑖. 
A recent paper [18] gives the proof of Theorem 3.5. The proof has verified the following: 

(𝑛 + 1)𝑚+1 − 1 = (
𝑚 + 1

𝑚
) 𝑆𝑚 + (

𝑚 + 1
𝑚 − 1

) 𝑆𝑚−1 + (
𝑚 + 1
𝑚 − 2

) 𝑆𝑚−2+. . . + (
𝑚 + 1

1
) 𝑆1 + 𝑛 

wherein, the 𝑆𝑚(𝑛) is simply written again as 𝑆𝑚. 

We present the next two theorems for the even and odd m in the 𝑛(𝑚)-factoriangular numbers. 

Theorem 3.6: The 𝑛(𝑚)-factoriangular number for even 𝑚 = 2𝑘 is given by the formula  

𝐹𝑡𝑛(𝑚) = 𝐹𝑡𝑛(2𝑘) = (𝑛!)2𝑘 +
2𝑛 + 1

2𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑇𝑛 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑇𝑛 = 1 + 2 + 3+. . . +𝑛 = 𝑛(𝑛 + 1)/2, and 𝑃(𝑛2𝑘−3) is a polynomial in 𝑛 of 

degree 2𝑘 − 3, for natural numbers 𝑛, 𝑘 ≥ 1. 
A recent paper [20] gives the proof of Theorem 3.6. The proof has verified the following: 

𝑆2(1)  =    𝑆2 =
2𝑛 + 1

3
𝑇𝑛 

𝑆2(2)  =    𝑆4 =
2𝑛 + 1

5
[𝑛2 + (𝑛 −

1

3
)]𝑇𝑛 

𝑆2(3)  =    𝑆6 =
2𝑛 + 1

7
[𝑛4 + (2𝑛3 − 𝑛 +

1

3
)]𝑇𝑛 

𝑆2(4)  =    𝑆8 =
2𝑛 + 1

9
[𝑛6 + (3𝑛5 + 𝑛4 − 3𝑛3 −

1

5
𝑛2 +

9

5
𝑛 −

3

5
)]𝑇𝑛 



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𝑆2𝑘 =
2𝑛 + 1

2𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑇𝑛 

wherein, the 𝑆𝑚(𝑛) is simply written again as 𝑆𝑚 (e.g., 𝑆2 means 𝑆2(𝑛)). 

Theorem 3.7: The 𝑛(𝑚)-factoriangular number for odd 𝑚 = 2𝑘 + 1 is given by the formula 

𝐹𝑡𝑛(𝑚) = 𝐹𝑡𝑛(2𝑘+1) = (𝑛!)2𝑘+1 +
𝑛(𝑛 + 1)

𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑇𝑛 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑇𝑛 = 1 + 2 + 3+. . . +𝑛 = 𝑛(𝑛 + 1)/2, and 𝑃(𝑛2𝑘−3) is a polynomial in 𝑛 of 

degree 2𝑘 − 3, for natural numbers 𝑛, 𝑘 ≥ 1. 
A recent paper [20] gives the proof of Theorem 3.7. The proof has verified the following: 

𝑆2(1)+1  =   𝑆3 =
𝑛(𝑛 + 1)

2
𝑇𝑛 

𝑆2(2)+1  =   𝑆5 =
𝑛(𝑛 + 1)

3
[𝑛2 + (𝑛 −

1

2
)]𝑇𝑛 

𝑆2(3)+1  =   𝑆7 =
𝑛(𝑛 + 1)

4
[𝑛4 + (2𝑛3 −

1

3
𝑛2 −

4

3
𝑛 +

2

3
)]𝑇𝑛 

𝑆2(4)+1  =   𝑆9 =
𝑛(𝑛 + 1)

5
[𝑛6 + (3𝑛5 +

1

2
𝑛4 − 4𝑛3 +

1

2
𝑛2 + 3𝑛 −

3

2
)]𝑇𝑛 

𝑆2𝑘+1 =
𝑛(𝑛 + 1)

𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑇𝑛 

wherein, the 𝑆𝑚(𝑛) is simply written again as 𝑆𝑚 (e.g., 𝑆3 means 𝑆3(𝑛)). 

We also present the next two theorems for the even and odd m in the 𝑛(𝑚)-factoriangular numbers 
that involves representation in the sum of powers instead of the triangular numbers. 

Theorem 3.8: The 𝑛(𝑚)-factoriangular number for even 𝑚 = 2𝑘 is given by the formula  

𝐹𝑡𝑛(𝑚) = 𝐹𝑡𝑛(2𝑘) = (𝑛!)2𝑘 +
3

2𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑆2(𝑛) 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑆2(𝑛) = 𝑛(𝑛 + 1)(2𝑛 + 1)/6, 𝑆3(𝑛) = 𝑛2(𝑛 + 1)2/4, and 𝑃(𝑛2𝑘−3) is a 

polynomial in 𝑛 of degree 2𝑘 − 3, for natural numbers 𝑛 ≥ 1and 𝑘 > 1. 
A recent paper [20] gives the proof of Theorem 3.8. The proof has verified the following: 

𝑆2(2)  =    𝑆4 =
3

5
[𝑛2 + (𝑛 −

1

3
)]𝑆2 

𝑆2(3)  =    𝑆6 =
3

7
[𝑛4 + (2𝑛3 − 𝑛 +

1

3
)]𝑆2 

𝑆2(4)  =    𝑆8 =
3

9
[𝑛6 + (3𝑛5 + 𝑛4 − 3𝑛3 −

1

5
𝑛2 +

9

5
𝑛 −

3

5
)]𝑆2 

𝑆2𝑘 =
3

2𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑆2 

Where in, the 𝑆𝑚(𝑛) is simply written again as 𝑆𝑚 (e.g., 𝑆4 means 𝑆4(𝑛)). 

Theorem 3.9: The 𝑛(𝑚)-factoriangular number for odd 𝑚 = 2𝑘 + 1 is given by the formula 

𝐹𝑡𝑛(𝑚) = 𝐹𝑡𝑛(2𝑘+1) = (𝑛!)2𝑘+1 +
2

𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑆3(𝑛) 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑆2(𝑛) = 𝑛(𝑛 + 1)(2𝑛 + 1)/6, 𝑆3(𝑛) = 𝑛2(𝑛 + 1)2/4, and 𝑃(𝑛2𝑘−3) is a 

polynomial in 𝑛 of degree 2𝑘 − 3, for natural numbers 𝑛 ≥ 1and 𝑘 > 1. 
A recent paper [20] gives the proof of Theorem 3.9. The proof has verified the following: 

𝑆2(2)+1  =   𝑆5 =
2

3
[𝑛2 + (𝑛 −

1

2
)]𝑆3 

𝑆2(3)+1  =   𝑆7 =
2

4
[𝑛4 + (2𝑛3 −

1

3
𝑛2 −

4

3
𝑛 +

2

3
)]𝑆3 



897 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 9, No. 3: 891-904, 2025 
DOI: 10.55214/25768484.v9i3.5378 
© 2025 by the author; licensee Learning Gate 

 

𝑆2(4)+1  =   𝑆8 =
2

5
[𝑛6 + (3𝑛5 +

1

2
𝑛4 − 4𝑛3 +

1

2
𝑛2 + 3𝑛 −

3

2
)]𝑆3 

𝑆2𝑘+1 =
2

𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑆3 

Where in, the 𝑆𝑚(𝑛) is simply written again as 𝑆𝑚 (e.g., 𝑆5 means 𝑆5(𝑛)). 
The {𝐹𝑡𝑛,𝑘} and {𝐹𝑡𝑛(𝑚)} can be combined to produce another generalization of the sequence of n-

factoriangular numbers, which is the sequence } for natural numbers 𝑛, 𝑘, 𝑚 ≥ 1 that follow the form 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (1𝑚 ⋅ 2𝑚 ⋅ 3𝑚 ⋅⋅⋅ 𝑛𝑚) + (1𝑚 + 2𝑚 + 3𝑚+. . . +𝑘𝑚) 

We call the numbers in this sequence as (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers and define as follows: 

Definition 3.10: The (𝑛(𝑚), 𝑘(𝑚))-factoriangular number is defined by the formula 

 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (𝑛!)𝑚 + 𝑆𝑚(𝑘) 

where (𝑛!)𝑚 = 1𝑚 ⋅ 2𝑚 ⋅ 3𝑚 ⋅⋅⋅ 𝑛𝑚 and 𝑆𝑚(𝑘) = 1𝑚 + 2𝑚 + 3𝑚+. . . +𝑘𝑚 for natural numbers 

𝑛, 𝑘, 𝑚 ≥ 1. 

From the above definitions, when 𝑚 = 1, the (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers are the same as 

the (𝑛, 𝑘)-factoriangular numbers or 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛,𝑘; when 𝑛 = 𝑘, the (𝑛(𝑚), 𝑘(𝑚))-factoriangular 

numbers are the same as the 𝑛(𝑚)-factoriangular numbers or 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(𝑚); and 𝑚 = 1 and 𝑛 =

𝑘, the (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers are the same as the n-factoriangular numbers or 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) =  𝐹𝑡𝑛. 

The sequence of (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers can be further generalized into the sequence 

{𝐹𝑡𝑛(𝑎),𝑘(𝑏)} for natural numbers 𝑛, 𝑘, 𝑎, 𝑏 ≥ 1, which follow the form 

𝐹𝑡𝑛(𝑎),𝑘(𝑏) = (1𝑎 ⋅ 2𝑎 ⋅ 3𝑎 ⋅⋅⋅ 𝑛𝑎) + (1𝑏 + 2𝑏 + 3𝑏+. . . +𝑘𝑏) 

We call the numbers in this sequence as (𝑛(𝑎), 𝑘(𝑏))-factoriangular numbers and define as follows: 

Definition 3.11: The (𝑛(𝑎), 𝑘(𝑏))-factoriangular number is defined by the formula 

 𝐹𝑡𝑛(𝑎),𝑘(𝑏) = (𝑛!)𝑎 + 𝑆𝑏(𝑘) 

where (𝑛!)𝑎 = 1𝑎 ⋅ 2𝑎 ⋅ 3𝑎 ⋅⋅⋅ 𝑛𝑎 and 𝑆𝑏(𝑘) = 1𝑏 + 2𝑏 + 3𝑏+. . . +𝑘𝑏 for natural numbers 𝑛, 𝑘, 𝑎, 𝑏 ≥ 1. 

From the above definitions, when 𝑎 = 𝑏 = 𝑚, the (𝑛(𝑎), 𝑘(𝑏))-factoriangular numbers are the same 

as the (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers or 𝐹𝑡𝑛(𝑎),𝑘(𝑏) = 𝐹𝑡𝑛(𝑚),𝑘(𝑚); when 𝑎 = 𝑏 = 1, the 

(𝑛(𝑎), 𝑘(𝑏))-factoriangular numbers are the same as the (𝑛, 𝑘)-factoriangular numbers or 𝐹𝑡𝑛(𝑎),𝑘(𝑏) =

𝐹𝑡𝑛,𝑘; when 𝑛 = 𝑘 and 𝑎 = 𝑏 = 𝑚, the (𝑛(𝑎), 𝑘(𝑏))-factoriangular numbers are the same as the 𝑛(𝑚)-

factoriangular numbers or 𝐹𝑡𝑛(𝑎),𝑘(𝑏) = 𝐹𝑡𝑛(𝑚) ; and when 𝑛 = 𝑘 and 𝑎 = 𝑏 = 1, the (𝑛(𝑎), 𝑘(𝑏))-

factoriangular numbers are the same as the n-factoriangular numbers or 𝐹𝑡𝑛(𝑎),𝑘(𝑏) = 𝐹𝑡𝑛.   

The above definitions were taken from previous study Castillo [20]. However, the said study 

focuses only on the 𝑛(𝑚)-factoriangular numbers. In the present study, we focus on the (𝑛(𝑚), 𝑘(𝑚))-

factoriangular numbers and on the (𝑛(𝑎), 𝑘(𝑏))-factoriangular numbers.   
 
3.2. Main Results 

3.2.1. On the (𝑛(𝑚), 𝑘(𝑚))-Factoriangular Numbers 

We integrate the notions of (𝑛, 𝑘)-factoriangular numbers and 𝑛(𝑚)-factoriangular numbers to form 

the (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers (see Definition 3.10). We now prove the succeeding theorems 

and give examples of sequences of (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers. For simplicity of notation and 

to avoid confusion between a function and a multiplication, we use 𝑆𝑚 in lieu of 𝑆𝑚(𝑘) in the proofs.  

Theorem 3.12: For natural numbers 𝑛, 𝑘, 𝑚 ≥ 1, the (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers can be 
determined by the formula  



898 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 9, No. 3: 891-904, 2025 
DOI: 10.55214/25768484.v9i3.5378 
© 2025 by the author; licensee Learning Gate 

 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (𝑛!)𝑚 +
1

𝑚 + 1
[(𝑘 + 1)[(𝑘 + 1)𝑚 − 1] − ∑ (

𝑚 + 1
𝑖

)

𝑚−1

𝑖=1

𝑆𝑖(𝑘)] 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛 and 𝑆𝑖(𝑘) = 1𝑖 + 2𝑖 + 3𝑖+. . . +𝑘𝑖.  

Proof: From Definition 3.10, 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (𝑛!)𝑚 + 𝑆𝑚(𝑘). What we need to show is that 

𝑆𝑚(𝑘) =
1

𝑚 + 1
[(𝑘 + 1)[(𝑘 + 1)𝑚 − 1] − ∑ (

𝑚 + 1
𝑖

)

𝑚−1

𝑖=1

𝑆𝑖(𝑘)] 

From a previous paper [20] we deduce that 

(𝑘 + 1)𝑚+1 − 1 = (
𝑚 + 1

𝑚
) 𝑆𝑚 + (

𝑚 + 1
𝑚 − 1

) 𝑆𝑚−1 + (
𝑚 + 1
𝑚 − 2

) 𝑆𝑚−2+. . . + (
𝑚 + 1

1
) 𝑆1 + 𝑘 

or 

(𝑘 + 1)𝑚+1 − (𝑘 + 1) = (𝑚 + 1)𝑆𝑚 + (
𝑚 + 1

1
) 𝑆1 + (

𝑚 + 1
2

) 𝑆2+. . . + (
𝑚 + 1
𝑚 − 1

) 𝑆𝑚−1 

and then, 

𝑆𝑚 =
1

𝑚 + 1
[(𝑘 + 1)[(𝑘 + 1)𝑚 − 1] − ∑ (

𝑚 + 1
𝑖

)

𝑚−1

𝑖=1

𝑆𝑖] 

It follows shortly that 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (𝑛!)𝑚 +
1

𝑚 + 1
[(𝑘 + 1)[(𝑘 + 1)𝑚 − 1] − ∑ (

𝑚 + 1
𝑖

)

𝑚−1

𝑖=1

𝑆𝑖(𝑘)] 

 

Theorem 3.13: The (𝑛(𝑚), 𝑘(𝑚))-factoriangular number for even 𝑚 = 2𝑗 is given by the formula  

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗),𝑘(2𝑗) = (𝑛!)2𝑗 +
2𝑘 + 1

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑇𝑘 = 1 + 2 + 3+. . . +𝑘 = 𝑘(𝑘 + 1)/2, and 𝑃(𝑘2𝑗−3) is a polynomial in 𝑘 of 

degree 2𝑗 − 3, for natural numbers 𝑛, 𝑘, 𝑗 ≥ 1.  

Proof: From Definition 3.10, 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (𝑛!)𝑚 + 𝑆𝑚(𝑘). We need to show that  

𝑆𝑚(𝑘) = 𝑆2𝑗(𝑘) =
2𝑘 + 1

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

It is similar to a previous proof [20] that 

𝑆2𝑘(𝑛) =
2𝑛 + 1

2𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑇𝑛 

we simply have 

𝑆2𝑗(𝑘) =
2𝑘 + 1

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

and then 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗),𝑘(2𝑗) = (𝑛!)2𝑗 +
2𝑘 + 1

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

 

Theorem 3.14: The (𝑛(𝑚), 𝑘(𝑚))-factoriangular number for odd 𝑚 = 2𝑗 + 1 is given by the formula 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1) = (𝑛!)2𝑗+1 +
𝑘(𝑘 + 1)

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑇𝑘 = 1 + 2 + 3+. . . +𝑘 = 𝑘(𝑘 + 1)/2, and 𝑃(𝑘2𝑗−3) is a polynomial in 𝑘 of 

degree 2𝑗 − 3, for natural numbers 𝑛, 𝑘, 𝑗 ≥ 1.  

Proof: From Definition 3.10, 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (𝑛!)𝑚 + 𝑆𝑚(𝑘). We need to show that  



899 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 9, No. 3: 891-904, 2025 
DOI: 10.55214/25768484.v9i3.5378 
© 2025 by the author; licensee Learning Gate 

 

𝑆𝑚(𝑘) = 𝑆2𝑗+1(𝑘) =
𝑘(𝑘 + 1)

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

It is similar to a previous proof [20] that  

𝑆2𝑘+1(𝑛) =
𝑛(𝑛 + 1)

𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑇𝑛 

we also have 

𝑆2𝑗+1(𝑘) =
𝑘(𝑘 + 1)

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

Hence, 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1) = (𝑛!)2𝑗+1 +
𝑘(𝑘 + 1)

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

 

Theorem 3.15: The (𝑛(𝑚), 𝑘(𝑚))-factoriangular number for even 𝑚 = 2𝑗 is given by the formula  

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗),𝑘(2𝑗) = (𝑛!)2𝑗 +
3

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆2(𝑘) 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑆2(𝑘) = 𝑘(𝑘 + 1)(2𝑘 + 1)/6, and 𝑃(𝑘2𝑗−3) is a polynomial in 𝑘 of degree 

2𝑗 − 3, for natural numbers 𝑛, 𝑘 ≥ 1and 𝑗 > 1.  

Proof: From Definition 3.10, 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (𝑛!)𝑚 + 𝑆𝑚(𝑘). We need to show that  

𝑆𝑚(𝑘) = 𝑆2𝑗(𝑘) =  
3

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆2(𝑘) 

It is similar to a previous proof [20] that 

𝑆2𝑘(𝑛) =
3

2𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑆2(𝑛) 

we have 

𝑆2𝑗(𝑘) =  
3

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆2(𝑘) 

and then 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗),𝑘(2𝑗) = (𝑛!)2𝑗 +
3

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆2(𝑘) 

 

Theorem 3.16: The (𝑛(𝑚), 𝑘(𝑚))-factoriangular number for odd 𝑚 = 2𝑗 + 1 is given by the formula 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1) = (𝑛!)2𝑗+1 +
2

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆3(𝑘) 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑆3(𝑘) = 𝑘2(𝑘 + 1)2/4, and 𝑃(𝑘2𝑗−3) is a polynomial in 𝑘 of degree 2𝑗 − 3, 

for natural numbers 𝑛, 𝑘 ≥ 1and 𝑗 > 1.  

Proof: From Definition 3.10, 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (𝑛!)𝑚 + 𝑆𝑚(𝑘). We need to show that  

𝑆𝑚(𝑘) = 𝑆2𝑗+1(𝑘) =
2

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆3(𝑘) 

It is similar to a previous proof [20] that 

𝑆2𝑘+1(𝑚) =
2

𝑘 + 1
[𝑛2𝑘−2 + 𝑃(𝑛2𝑘−3)]𝑆3(𝑚) 

we have 

𝑆2𝑗+1(𝑘) =
2

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆3(𝑘) 

Thus,  



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𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1) = (𝑛!)2𝑗+1 +
2

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆3(𝑘) 

Two examples of 𝐹𝑡𝑛(𝑚),𝑘(𝑚) , with 𝑚 > 1, are presented here. The first example is when 𝑚 = 2 and 

the (𝑛(2), 𝑘(2))-factoriangular numbers are given in Table 2. 
 
Table 2.  

Table of (n(2), k(2))-factoriangular numbers. 

𝒏 \ 𝒌 1 2 3 4 5  𝒌 
1 2 6 15 31 56  1 + 𝑆2(𝑘) 
2 5 9 18 34 59  4 + 𝑆2(𝑘) 
3 37 41 50 66 91  36 + 𝑆2(𝑘) 
4 577 581 590 606 631  576 + 𝑆2(𝑘) 
5 14401 14405 14414 14430 14455  14400 + 𝑆2(𝑘) 

        

𝑛 (𝑛!)2 + 1 (𝑛!)2 + 5 (𝑛!)2 + 14 (𝑛!)2 + 30 (𝑛!)2 + 55  (𝑛!)2 + 𝑆2(𝑘) 
 

Then, the sequence of (𝑛(2), 𝑘(2))-factoriangular numbers is given by 

{𝐹𝑡𝑛(2),𝑘(2)} = {2, 5, 6, 9, 37, 15, 41, 18, 50, . . . } 

for (n,k) = (1,1), (2,1), (1,2), (2,2), (3,1), (1,3), (3,2), (2,3), (3,3), … . 

The second example is when 𝑚 = 3 and the (𝑛(3), 𝑘(3))-factoriangular numbers are given in Table 
3. 
 
Table 3.  

Table of (n(3), k(3))-factoriangular numbers. 

𝒏 \ 𝒌 1 2 3 4  𝒌 
1 2 10 37 101  1 + 𝑆3(𝑘) 
2 9 17 44 108  8 + 𝑆3(𝑘) 
3 217 225 252 316  216 + 𝑆3(𝑘) 
4 13825 13833 13860 13924  13824 + 𝑆3(𝑘) 

       

𝒏 (𝑛!)3 + 1 (𝑛!)3 + 9 (𝑛!)3 + 36 (𝑛!)3 + 100  (𝑛!)3 + 𝑆3(𝑘) 

 

Then, the sequence of (𝑛(3), 𝑘(3))-factoriangular numbers is given by 

{𝐹𝑡𝑛(3),𝑘(3)} = {2, 9, 10, 17, 217, 37, 225, 44, 252, . . . } 

for (n,k) = (1,1), (2,1), (1,2), (2,2), (3,1), (1,3), (3,2), (2,3), (3,3), … . 

3.2.2. On the (𝑛(𝑎), 𝑘(𝑏))-Factoriangular Numbers 

We further generalized (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers to have the (𝑛(𝑎), 𝑘(𝑏))-factoriangular 
numbers (see Definition 3.11). We now present the following theorems whose proofs are similar to the 

previous theorems on (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers. We also give examples of sequences of 

(𝑛(𝑎), 𝑘(𝑏))-factoriangular numbers.  

Theorem 3.17: For natural numbers 𝑛, 𝑘, 𝑎, 𝑏 ≥ 1, the (𝑛(𝑎), 𝑘(𝑏))-factoriangular numbers can be 
determined by the formula  

𝐹𝑡𝑛(𝑎),𝑘(𝑏) = (𝑛!)𝑎 +
1

𝑏 + 1
[(𝑘 + 1)[(𝑘 + 1)𝑏 − 1] − ∑ (

𝑏 + 1
𝑖

)

𝑏−1

𝑖=1

𝑆𝑖(𝑘)] 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛 and 𝑆𝑖(𝑘) = 1𝑖 + 2𝑖 + 3𝑖+. . . +𝑘𝑖. 
The proof of Theorem 3.17 is similar to the proof of Theorem 3.12. 



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Theorem 3.18: The (𝑛(𝑎), 𝑘(𝑏))-factoriangular number for even 𝑏 = 2𝑗 is given by the formula  

𝐹𝑡𝑛(𝑎),𝑘(𝑏) = 𝐹𝑡𝑛(𝑎),𝑘(2𝑗) = (𝑛!)𝑎 +
2𝑘 + 1

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑇𝑘 = 1 + 2 + 3+. . . +𝑘 = 𝑘(𝑘 + 1)/2, and 𝑃(𝑘2𝑗−3) is a polynomial in 𝑘 of 

degree 2𝑗 − 3, for natural numbers 𝑛, 𝑘, 𝑗 ≥ 1.  
The proof of Theorem 3.18 is similar to the proof of Theorem 3.13. 

Theorem 3.19: The (𝑛(𝑎), 𝑘(𝑏))-factoriangular number for odd 𝑏 = 2𝑗 + 1 is given by the formula 

𝐹𝑡𝑛(𝑎),𝑘(𝑏) = 𝐹𝑡𝑛(𝑎),𝑘(2𝑗+1) = (𝑛!)𝑎 +
𝑘(𝑘 + 1)

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑇𝑘 = 1 + 2 + 3+. . . +𝑘 = 𝑘(𝑘 + 1)/2, and 𝑃(𝑘2𝑗−3) is a polynomial in 𝑘 of 

degree 2𝑗 − 3, for natural numbers 𝑛, 𝑘, 𝑗 ≥ 1.  
The proof of Theorem 3.19 is similar to the proof of Theorem 3.14. 

Theorem 3.20: The (𝑛(𝑎), 𝑘(𝑏))-factoriangular number for even 𝑏 = 2𝑗 is given by the formula  

𝐹𝑡𝑛(𝑎),𝑘(𝑏) = 𝐹𝑡𝑛(𝑎),𝑘(2𝑗) = (𝑛!)𝑎 +
3

2𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆2(𝑘) 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑆2(𝑘) = 𝑘(𝑘 + 1)(2𝑘 + 1)/6, and 𝑃(𝑘2𝑗−3) is a polynomial in 𝑘 of degree 

2𝑗 − 3, for natural numbers 𝑛, 𝑘 ≥ 1and 𝑗 > 1.  
The proof of Theorem 3.20 is similar to the proof of Theorem 3.15. 

Theorem 3.21: The (𝑛(𝑎), 𝑘(𝑏))-factoriangular number for odd 𝑏 = 2𝑗 + 1 is given by the formula 

𝐹𝑡𝑛(𝑎),𝑘(𝑏) = 𝐹𝑡𝑛(𝑎),𝑘(2𝑗+1) = (𝑛!)𝑎 +
2

𝑗 + 1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆3(𝑘) 

where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛, 𝑆3(𝑘) = 𝑘2(𝑘 + 1)2/4, and 𝑃(𝑘2𝑗−3) is a polynomial in 𝑘 of degree 2𝑗 − 3, 

for natural numbers 𝑛, 𝑘 ≥ 1and 𝑗 > 1.  
The proof of Theorem 3.21 is similar to the proof of Theorem 3.16. 

Three examples of 𝐹𝑡𝑛(𝑎),𝑘(𝑏), with 𝑎, 𝑏 ≥ 1 but not both 𝑎, 𝑏 = 1, are presented here. When 𝑎 = 1 

and 𝑏 = 2, the (𝑛(1), 𝑘(2))-factoriangular numbers are given in Table 4. 
 
Table 4.  

Table of (n(1), k(2))-factoriangular numbers. 

𝒏 \ 𝒌 1 2 3 4 5  𝒌 

1 2 6 15 31 56  1 + 𝑆2(𝑘) 
2 3 7 16 32 57  2 + 𝑆2(𝑘) 
3 7 11 20 36 61  6 + 𝑆2(𝑘) 
4 25 29 38 54 79  24 + 𝑆2(𝑘) 
5 121 125 134 150 175  120 + 𝑆2(𝑘) 

        

𝒏 𝑛! + 1 𝑛! + 5 𝑛! + 14 𝑛! + 30 𝑛! + 55  𝑛! + 𝑆2(𝑘) 

 

Then, the sequence of (𝑛(1), 𝑘(2))-factoriangular numbers is given by 

{𝐹𝑡𝑛(1),𝑘(2)} = {2, 3, 6, 7, 7, 15, 11, 16, 20, . . . } 

for (n,k) = (1,1), (2,1), (1,2), (2,2), (3,1), (1,3), (3,2), (2,3), (3,3), … . 

When 𝑎 = 2 and 𝑏 = 1, the (𝑛(2), 𝑘(1))-factoriangular numbers are given in Table 5. 
 
 
 
 
 



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Table 5.  

Table of (n(2), k(1))-factoriangular numbers. 

𝒏\𝒌 1 2 3 4 5  𝒌 
1 2 4 7 11 16  1 + 𝑆1(𝑘) 
2 5 7 10 14 19  4 + 𝑆1(𝑘) 
3 37 39 42 46 51  36 + 𝑆1(𝑘) 
4 577 579 582 586 591  576 + 𝑆1(𝑘) 
5 14401 14403 14406 14410 14415  14400 + 𝑆1(𝑘) 

        

𝒏 (𝑛!)2 + 1 (𝑛!)2 + 3 (𝑛!)2 + 6 (𝑛!)2 + 10 (𝑛!)2 + 15  (𝑛!)2 + 𝑆1(𝑘) 

 

Then, the sequence of (𝑛(2), 𝑘(1))-factoriangular numbers is given by 

{𝐹𝑡𝑛(2),𝑘(1)} = {2, 5, 4, 7, 37, 7, 39, 10, 42, . . . } 

for (n,k) = (1,1), (2,1), (1,2), (2,2), (3,1), (1,3), (3,2), (2,3), (3,3), … . 

When 𝑎 = 2 and 𝑏 = 3, the (𝑛(2), 𝑘(3))-factoriangular numbers are given in Table 6.   
 
Table 6.  

Table of (n(2), k(3))-factoriangular numbers 

𝒏\𝒌 1 2 3 4 5  𝒌 
1 2 10 37 101 226  1 + 𝑆3(𝑘) 

2 5 13 40 104 229  4 + 𝑆3(𝑘) 

3 37 45 72 136 261  36 + 𝑆3(𝑘) 

4 577 585 612 676 801  576 + 𝑆3(𝑘) 

5 14401 14409 14436 14500 14625  14400 + 𝑆3(𝑘) 

        

𝒏 (𝑛!)2 + 1 (𝑛!)2 + 9 (𝑛!)2 + 36 (𝑛!)2 + 100 (𝑛!)2 + 225  (𝑛!)2 + 𝑆3(𝑘) 

 

Then, the sequence of (𝑛(2), 𝑘(3))-factoriangular numbers is given by 

{𝐹𝑡𝑛(2),𝑘(3)} = {2, 5, 10, 13, 37, 37, 45, 40, 72, . . . } 

for (n,k) = (1,1), (2,1), (1,2), (2,2), (3,1), (1,3), (3,2), (2,3), (3,3), … . 
 

4. Conclusions 
Research on factoriangular numbers is still relatively new with its introduction to the number 

theory literature only in 2015. A factoriangular number is formed by adding a factorial and its additive 
analog, a triangular number. When a triangular number is added to its corresponding factorial, the 

result is an n-factoriangular number. The sequence {𝐹𝑡𝑛} = {2, 5, 12, 34, 135, 741, 5068, … } is the 
sequence of n-factoriangular numbers. The terms in the sequence can be generated by using the formula 

𝐹𝑡𝑛 = 𝑛!  +  𝑇𝑛, where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛 and 𝑇𝑛 = 1 + 2 + 3+. . . +𝑛. This sequence of n-factoriangular 

numbers can be generalized in several ways. One generalization is the sequence of (𝑛, 𝑘)-factoriangular 

numbers of the form 𝐹𝑡𝑛,𝑘 = 𝑛!  +  𝑇𝑘, where 𝑛! = 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ 𝑛 and 𝑇𝑘 = 1 + 2 + 3+. . . +𝑘. Another 

generalization is the sequence of 𝑛(𝑚)-factoriangular numbers of the form 𝐹𝑡𝑛(𝑚) = (𝑛!)𝑚 + 𝑆𝑚(𝑛), 

where (𝑛!)𝑚 = 1𝑚 ⋅ 2𝑚 ⋅ 3𝑚 ⋅⋅⋅ 𝑛𝑚 and 𝑆𝑚(𝑛) = 1𝑚 + 2𝑚 + 3𝑚+. . . +𝑛𝑚. 

More generalizations can be made by integrating the concepts of (𝑛, 𝑘)-factoriangular numbers and 

𝑛(𝑚)-factoriangular numbers to produce the sequence of (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers of the 

form 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = (𝑛!)𝑚 + 𝑆𝑚(𝑘), where (𝑛!)𝑚 = 1𝑚 ⋅ 2𝑚 ⋅ 3𝑚 ⋅⋅⋅ 𝑛𝑚 and 𝑆𝑚(𝑘) = 1𝑚 + 2𝑚 +

3𝑚+. . . +𝑘𝑚. The sequence of (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers can be further generalized into the 

sequence of (𝑛(𝑎), 𝑘(𝑏))-factoriangular numbers of the form 𝐹𝑡𝑛(𝑎),𝑘(𝑏) = (𝑛!)𝑎 + 𝑆𝑏(𝑘), where (𝑛!)𝑎 =

1𝑎 ⋅ 2𝑎 ⋅ 3𝑎 ⋅⋅⋅ 𝑛𝑎 and 𝑆𝑏(𝑘) = 1𝑏 + 2𝑏 + 3𝑏+. . . +𝑘𝑏. 



903 

 

 

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The (𝑛(𝑚), 𝑘(𝑚))-factoriangular numbers can also be determined by the formula 𝐹𝑡𝑛(𝑚),𝑘(𝑚) =

(𝑛!)𝑚 +
1

𝑚+1
[(𝑘 + 1)[(𝑘 + 1)𝑚 − 1] − ∑ (

𝑚 + 1
𝑖

)𝑚−1
𝑖=1 𝑆𝑖(𝑘)] and the (𝑛(𝑎), 𝑘(𝑏))-factoriangular 

numbers by the formula 𝐹𝑡𝑛(𝑎),𝑘(𝑏) = (𝑛!)𝑎 +
1

𝑏+1
[(𝑘 + 1)[(𝑘 + 1)𝑏 − 1] − ∑ (

𝑏 + 1
𝑖

)𝑏−1
𝑖=1 𝑆𝑖(𝑘)]. The 

formulas for (𝑛(𝑚), 𝑘(𝑚))-factoriangular number for even 𝑚 = 2𝑗 are 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗),𝑘(2𝑗) =

(𝑛!)2𝑗 +
2𝑘+1

2𝑗+1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 and 𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗),𝑘(2𝑗) = (𝑛!)2𝑗 +

3

2𝑗+1
[𝑘2𝑗−2 +

𝑃(𝑘2𝑗−3)]𝑆2(𝑘). The formulas for (𝑛(𝑚), 𝑘(𝑚))-factoriangular number for odd 𝑚 = 2𝑗 + 1 are 

𝐹𝑡𝑛(𝑚),𝑘(𝑚) = 𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1) = (𝑛!)2𝑗+1 +
𝑘(𝑘+1)

𝑗+1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑇𝑘 and 𝐹𝑡𝑛(𝑚),𝑘(𝑚) =

𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1) = (𝑛!)2𝑗+1 +
2

𝑗+1
[𝑘2𝑗−2 + 𝑃(𝑘2𝑗−3)]𝑆3(𝑘). Similar formulas can be provided for  

(𝑛(𝑎), 𝑘(𝑏))-factoriangular number for even 𝑚 = 2𝑗 and for odd 𝑚 = 2𝑗 + 1.  
Triangular arrays of factoriangular numbers may be formed from the tables of generalized 

factoriangular numbers. This will be of future interest to other mathematical explorers, especially those 
in the field of recreational mathematics.  
 

Transparency:  
The author confirms that the manuscript is an honest, accurate,  and  transparent  account  of  the  
study; that  no  vital  features  of  the  study  have  been  omitted;  and  that  any  discrepancies  from  
the  study  as planned have been explained. This study followed all ethical practices during writing. 
 

Copyright: 
© 2025 by the authors. This open-access article is distributed under the terms and conditions of the 
Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). 
 

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