The Intellectualism.Org Question Of The Day for
October 02 , 2003:

Is there any pattern to the set of trapezoidal numbers?


Trapezoidal numbers are those natural numbers which result from the difference of two triangular numbers differing by at least 1 row.

As an example, the lowest triangular number is 5; the difference of triangular numbers 6 and 1. The next highest triangular number is 7, which is the difference of the triangular numbers 10 and 3.

Mathematically, trapezoidal numbers are formed by a sum of a continuous range of natural numbers, with the lowest term in the sum greater than 1. There are, of course, an infinite number of them; the proof is trivial -- there are an infinite number of triangular numbers, since there are an infinite number of positive integers -- and for every triangular number N greater than 3, there is a trapezoidal number N-1 (among many others; there are *far* more trapezoidal numbers for any given finite row limit than there are triangular numbers.

There is a pattern to them. Each trapezoidal number construction is a member of a particular series uniquely identified by its highest-numbered row. For instance, the first series contains just the number 5, since this is the only construction possible starting at row 3. Each successive series contains one more member -- for instance, starting at row 4, there are the trapezoidal numbers 7=4+3 and 9=4+3+2 . The pattern is similar to the pattern made by the composite numbers, but since the series do not start in the same location, it differs.

Here is a chart showing the patterns in the first several trapezoidal numbers. Red represents trapezoidal numbers with two rows, Orange with three rows, etc. The numbers along the bottom show how many trapezoidal constructions, if any, are possible for each number.

Powers of 2 seem to never have a trapezoidal construction, but this is a Question for another day.

 


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