On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers

A trapezoidal number, a sum of at least two consecutive positive integers, is a figurate number that can be represented by points rearranged in the plane as a trapezoid. Such numbers have been of interest and extensively studied. In this paper, a generalization of trapezoidal numbers has been introduced. For each positive integerm, a positive integerN is called anm-trapezoidal number if N can be written as an arithmetic series of at least 2 terms with common difference m. Properties of m-trapezoidal numbers have been studied together with their trapezoidal representations. In the special case wherem = 2, the characterization and enumeration of such numbers have been given as well as illustrative examples. Precisely, for a fixed 2-trapezoidal number N, the ways and the number of ways to writeN as an arithmetic series with common difference 2 have been determined. Some remarks on 3-trapezoidal numbers have been provided as well.


Introduction
A triangular number is a figurate number that can be represented by an equilateral triangular arrangement of points equally spaced.For each positive integer ℓ, the ℓth triangular number is the number of points composing a triangle with ℓ points on a side and is equal to the sum of the ℓ natural numbers of the form Tri(ℓ) = 1 + 2 + 3 + ⋅ ⋅ ⋅ + ℓ.The ℓth triangular number can be represented as points in an equilateral triangle as in Figure 1.
Triangular numbers have been studied since the ancient Greeks.The Pythagoreans revered the Tetractys which is Tri (4).Triangular numbers have applications to other areas of number theory, such as perfect numbers and binomial coefficients.They are also practically the simplest example of an arithmetic sequence.Therefore, the triangular numbers have fascinated people and cultures all over the world (see [1][2][3] and references therein).
From the definition, it is not difficult to see that every trapezoidal number  can be represented by a rearrangement of  points in the plane as a trapezoid as in Figure 2.For convenience, denote by (ℓ, ) the number of the form (1). The characterization and enumeration of trapezoidal numbers have been given in [4,8].The main results are summarized as follows.
Theorem 1 ([4, Proposition 1]).Let  be a positive integer.Then  is a trapezoidal number if and only if  is not of the form 2  for all  ∈ N ∪ {0}.
be a positive integer such that  ≥ 1 and  ≥ 0 are integers,  1 ,  2 , . . .,   are distinct odd primes, and   ≥ 1 is an integer for all  = 1, 2, . . ., .Then  is a trapezoidal number and there are ways of writing  as a sum of at least two consecutive integers.

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International Journal of Mathematics and Mathematical Sciences . . .d The ℓth triangular number.Some properties of nontrapezoidal numbers can be found in [9].
Triangular numbers and trapezoidal numbers have a closed connection (see Section 2 for more details) with a rectangular number which is defined to be where ℓ ≥ 2 and  ≥ 2 are integers.A rectangular number can be represented as a rectangle as in Figure 3.
In this paper, we focus on a general concept of trapezoidal numbers.For each positive integer , a positive integer  is called an -trapezoidal number if  can be written as an arithmetic series of at least 2 terms with common difference .It follows that an -trapezoidal number can be represented as for some integers  ≥ 0 and ℓ ≥ 2. It is not difficult to see that a 1-trapezoidal number is a classical trapezoidal number.For convenience, denote by (ℓ, , ) the series in (4).We note that an -trapezoidal number is not uniquely determined by a triple (ℓ, , ) (see Example 3).Every trapezoidal number can be represented by an arrangement of points in the plane as a trapezoid.Some examples are given as follows.( The above series can be represented as trapezoids of 18 points in the plane as in Figure 4.In this paper, we focus on properties of -trapezoidal numbers and their representations as trapezoids in the plane.The characterization and enumeration of -trapezoidal numbers are studied in the special case where  = 2.The paper is organized as follows.In Section 2, general properties of -trapezoidal numbers are discussed as well as links with other figurate numbers.In Section 3, the characterization and enumeration of 2-trapezoidal numbers have been given together with some illustrative examples.Remarks on 3trapezoidal numbers have been provided in Section 4. Conclusion and open problems are given in Section 5.

Generalized Trapezoidal Numbers
In this section, we focus on general properties of trapezoidal numbers and links with other figurate numbers such as triangular numbers, trapezoidal number, and rectangular numbers.
First, we simplify the formula for an -trapezoidal number.
From the formula in Lemma 5, the following properties can be deduced.Corollary 6.Let  be a positive integer.If  is even, then an -trapezoidal number (ℓ, , ) is a rectangular number for all integers  ≥ 0 and ℓ ≥ 2.

Corollary 7.
Let  be a positive integer.If  is odd, then an -trapezoidal number (ℓ, , ) is a rectangular number for all integers  ≥ 0 and ℓ ≥ 3.
Proof.Assume that  is odd.Let  ≥ 0 and ℓ ≥ 3 be integers.We consider the following two cases.
Hence, the result follows.
The next corollary follows immediately from Theorem 8.
The above relations can be represented in the plane as in Figures 6 and 7.
Theorem 11.Let  and  be positive integers.If  is an trapezoidal number such that  ≥  + 4, then  can be written as a sum of a rectangular number and an -trapezoidal number.
Proof.Assume that  is an -trapezoidal number such that  ≥  + 4. Then for some integers  ≥ 1 and  ≥ 2.

Characterization and Enumeration of 2-Trapezoidal Numbers
In this section, we focus on the special case where  = 2.The characterization and enumeration of 2-trapezoidal numbers are given together with some illustrative examples.The characterization of 2-trapezoidal numbers is given in the next theorem which is totally different from the case of 1-trapezoidal numbers in Theorem 1.
Theorem 13.Let  ≥ 2 be an integer.Then  is a 2-trapezoidal number if and only if  is not a prime.
Proof.Assume that  is a 2-trapezoidal number.By Corollary 6,  is a rectangular number.Hence,  is not a prime number.
From the proof of Theorem 13, a 2-trapezoidal number  can be represented as a series (ℓ, , 2) via the following steps: (1) Determine the divisors  of  such that 1 <  ≤ √ .

Let us consider the following examples.
Example 14.Consider the 2-trapezoidal number  = 54.We have 54 = 2 ⋅ 3 3 which can be written as arithmetic series of at least 2 terms with common difference 2 as in Table 1.Then 100 = 2 2 ⋅ 5 2 which can be written as arithmetic series of at least 2 terms with common difference 2 as in Table 2.
Then 175 = 5 2 ⋅ 7 which can be written as arithmetic series of at least 2 terms with common difference 2 as in Table 3.
By the definition, every 2-trapezoidal number can be written as an arithmetic series with common difference 2. In the following theorem, we determine the number of ways to write a 2-trapezoidal number in terms of an arithmetic series of at least 2 terms with common difference 2.
Theorem 17.Let  be a 2-trapezoidal number.Then the number of ways to write  as an arithmetic series of at least 2 terms with common difference 2 is where () is the number of divisors of .
Proof.From the proof of Theorem 13, it follows that  =  for some integers 1 <  ≤ .Next, we consider the following two cases.
Case 2 ( is not a square).In this case, we have 1 <  < .
Then the number of ways to write  as an arithmetic series of at least 2 terms with common difference 2 is the number of divisors  of  such that 1 <  < √ .Since  is not a square,  = 2 ⋅ ⋅ ⋅     , where  ≥ 1,  1 ,  2 , . . .,   are distinct odd primes and   is a positive integer for all  = 1, 2, . . .,  such that   is odd for some 1 ≤  ≤ .Then the number of divisors of  is () = ( 1 + 1)( 2 + 1) ⋅ ⋅ ⋅ (  + 1) which is even.Therefore, the number of ways to write  as an arithmetic series of at least 2 terms with common difference 2 is ()/2 − 1.
From the two cases, the result follows.
The next corollary is a direct consequence of Theorem 17.
Corollary 18.Let  be a 2-trapezoidal number.Then  has a unique representation as an arithmetic series of at least 2 terms with common difference 2 if and only one of the following statements holds: (1)  is a product of two distinct primes.
(2)  is the square of a prime.
(3)  is the cube of a prime.
Some illustrative examples of the number of ways to write a 2-trapezoidal number  as an arithmetic series of at least 2 terms with common difference 2 are shown in Tables 4 and 5.

Some Properties of 3-Trapezoidal Numbers
In this section, we focus on properties of 3-trapezoidal numbers.A necessary condition for a positive integer to be a 3-trapezoidal number is given.However, this condition is not sufficient.
Theorem 19.Let  be a positive integer.If  is a 3-trapezoidal number, then  is not in the form of 2  for all  ∈ N ∪ {0}.for some ℓ ≥ 2 and  ≥ 0. We consider the following two cases.
We note that the necessary condition given in Theorem 19 is not sufficient.It is not difficult to see that 6 is not of the form 2  for all  ∈ N ∪ {0} but 6 is not a 3-trapezoidal number.

Conclusion and Remarks
A general concept of trapezoidal numbers has been introduced.Some properties of -trapezoidal numbers have been determined as well as links with other figurate numbers.Complete characterization and enumeration of 2-trapezoidal numbers are given.A necessary condition of a positive integer to be a 3-trapezoidal number is determined.However, the given condition is not sufficient.
In general, it is interesting to study the characterization and enumeration of -trapezoidal numbers with  ≥ 3.

Table 4 :
Number of representations of square 2-trapezoidal numbers.