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 integerm, a positive integerNis called anm-trapezoidal numberifNcan be written as an arithmetic series of at least2terms with common differencem. Properties ofm-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 fixed2-trapezoidal numberN, the ways and the number of ways to writeNas an arithmetic series with common difference2have been determined. Some remarks on3-trapezoidal numbers have been provided as well.
Highlights
For each positive integer m, a positive integer N is called an m-trapezoidal number if N can be written as an arithmetic series of at least 2 terms with common difference m
A triangular number is a figurate number that can be represented by an equilateral triangular arrangement of points 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
Summary
A triangular number is a figurate number that can be represented by an equilateral triangular arrangement of points spaced. 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. For each positive integer m, a positive integer N is called an m-trapezoidal number if N can be written as an arithmetic series of at least 2 terms with common difference m It follows that an m-trapezoidal number can be represented as (k + 1) + (k + 1 + m) + (k + 1 + 2m) + ⋅ ⋅ ⋅ (4). The positive integer 18 is a (1-)trapezoidal number represented in the forms of series.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
