Abstract
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 integer , a positive integer is called an -trapezoidal number if can be written as an arithmetic series of at least terms with common difference . Properties of -trapezoidal numbers have been studied together with their trapezoidal representations. In the special case where , the characterization and enumeration of such numbers have been given as well as illustrative examples. Precisely, for a fixed -trapezoidal number , the ways and the number of ways to write as an arithmetic series with common difference have been determined. Some remarks on -trapezoidal numbers have been provided as well.
1. 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 , the th triangular number is the number of points composing a triangle with points on a side and is equal to the sum of the natural numbers of the form The th 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 . 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 [1–3] and references therein).
A trapezoidal number (see [4], e.g.) is a generalization of a triangular number defined to be a sum of at least two consecutive positive integers. Precisely, a positive integer is a trapezoidal number iffor some integers and . Trapezoidal numbers form an important class of figurate numbers that has extensively been studied (see [2–7]).
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 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 for all .
Theorem 2 ([4, Proposition 2]). Let be a positive integer such that and are integers, are distinct odd primes, and is an integer for all . Then is a trapezoidal number and there areways of writing as a sum of at least two consecutive integers.
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 bewhere and 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 terms with common difference . It follows that an -trapezoidal number can be represented asfor some integers and . It is not difficult to see that a -trapezoidal number is a classical trapezoidal number. For convenience, denote by the series in (4).
We note that an -trapezoidal number is not uniquely determined by a triple (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.
Example 3. The positive integer is a (-)trapezoidal number represented in the forms of seriesThe above series can be represented as trapezoids of points in the plane as in Figure 4.
Example 4. The numbers and are examples of -trapezoidal and -trapezoidal numbers, respectively. They can be represented as trapezoids in Figure 5.
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 . 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 -trapezoidal numbers have been given together with some illustrative examples. Remarks on -trapezoidal numbers have been provided in Section 4. Conclusion and open problems are given in Section 5.
2. 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.
Lemma 5. Let , , and be integers. Then
Proof. From the definition, we haveas desired.
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 is a rectangular number for all integers and .
Proof. Assume that is even. Let and be integers. Then Since and , is a rectangular number.
Corollary 7. Let be a positive integer. If is odd, then an -trapezoidal number is a rectangular number for all integers and .
Proof. Assume that is odd. Let and be integers. We consider the following two cases.
Case 1 ( is even). Then for some and Since and , is a rectangular number.
Case 2 ( is odd). Then for some and Since and , is a rectangular number.
From the two cases, is rectangular for all and .
Trapezoidal numbers, -trapezoidal numbers, rectangular numbers, and triangular numbers are linked via the following relations.
Theorem 8. Let be integers such that , , and . Thenfor all positive integers .
Proof. Let be a positive integer. ThenHence, the result follows.
The next corollary follows immediately from Theorem 8.
Corollary 9. Let be integers such that , and . Then the following statements hold:(1).(2).
Illustrative examples of results in Corollary 9 are given as follows.
Example 10. Let and . From Theorem 8, we have 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 , then can be written as a sum of a rectangular number and an -trapezoidal number.
Proof. Assume that is an -trapezoidal number such that . Thenfor some integers and .
Let and . Then is a rectangular number and is an -trapezoidal number. It follows that Hence, is a sum of a rectangular number and an -trapezoidal number.
Example 12. Let , and . Then which can be represented in the plane as in Figure 8.
3. Characterization and Enumeration of 2-Trapezoidal Numbers
In this section, we focus on the special case where . The characterization and enumeration of -trapezoidal numbers are given together with some illustrative examples.
The characterization of -trapezoidal numbers is given in the next theorem which is totally different from the case of -trapezoidal numbers in Theorem 1.
Theorem 13. Let be an integer. Then is a -trapezoidal number if and only if is not a prime.
Proof. Assume that is a -trapezoidal number. By Corollary 6, is a rectangular number. Hence, is not a prime number.
Conversely, assume that is not a prime number. Then there exist integers such that . Choose and . Then , , and Hence, is a -trapezoidal number as desired.
From the proof of Theorem 13, a -trapezoidal number can be represented as a series via the following steps:(1)Determine the divisors of such that .(2)For each , compute .(3)Write , where and .Let us consider the following examples.
Example 14. Consider the -trapezoidal number . We have which can be written as arithmetic series of at least terms with common difference as in Table 1.
Example 15. Consider the -trapezoidal number . Then which can be written as arithmetic series of at least terms with common difference as in Table 2.
Example 16. Consider the -trapezoidal number . Then which can be written as arithmetic series of at least terms with common difference as in Table 3.
By the definition, every -trapezoidal number can be written as an arithmetic series with common difference . In the following theorem, we determine the number of ways to write a -trapezoidal number in terms of an arithmetic series of at least terms with common difference .
Theorem 17. Let be a -trapezoidal number. Then the number of ways to write as an arithmetic series of at least terms with common difference is where is the number of divisors of .
Proof. From the proof of Theorem 13, it follows that for some integers . Next, we consider the following two cases.
Case 1 ( is a square). In this case, we have . Then the number of ways to write as an arithmetic series of at least terms with common difference is the number of divisors of such that . Since is a square, for some , are distinct odd primes, and is an even positive integer for all . Then the number of divisors of is which is odd. Hence, the number of ways to write as an arithmetic series of at least terms with common difference is
Case 2 ( is not a square). In this case, we have . Then the number of ways to write as an arithmetic series of at least terms with common difference is the number of divisors of such that . Since is not a square, , where , are distinct odd primes and is a positive integer for all such that is odd for some . Then the number of divisors of is which is even. Therefore, the number of ways to write as an arithmetic series of at least terms with common difference is .
From the two cases, the result follows.
The next corollary is a direct consequence of Theorem 17.
Corollary 18. Let be a -trapezoidal number. Then has a unique representation as an arithmetic series of at least terms with common difference 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 -trapezoidal number as an arithmetic series of at least terms with common difference are shown in Tables 4 and 5.
4. Some Properties of 3-Trapezoidal Numbers
In this section, we focus on properties of -trapezoidal numbers. A necessary condition for a positive integer to be a -trapezoidal number is given. However, this condition is not sufficient.
Theorem 19. Let be a positive integer. If is a -trapezoidal number, then is not in the form of for all .
Proof. Assume that is a -trapezoidal number. Thenfor some and . We consider the following two cases.
Case 1 ( is odd). It follows that is even and . It follows that is odd and . Hence, for all .
Case 2 ( is even). We have that is odd and is odd. Since , it follows that and . Hence, for all .
Altogether, we have that for all as desired.
We note that the necessary condition given in Theorem 19 is not sufficient. It is not difficult to see that is not of the form for all but is not a -trapezoidal number.
5. 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 -trapezoidal numbers are given. A necessary condition of a positive integer to be a -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 .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by the Thailand Research Fund and the Office of Higher Education Commission of Thailand under Research Grant MRG6080012.