Street Broad Scientific
Mathematics and Computer Science Research
Volume 1 | 2011-2012
Non-S-Figurate Numbers Peter Cheng & Vinit Ranjan & Kelly Zhang ABSTRACT The triangular numbers include 1, 3, 6, 10, 15 .... These are the figurate numbers corresponding to a triangle. The motivation behind our research was to discover a formula that would output the non-figurate numbers, the natural numbers remaining after removing the figurate numbers corresponding to a certain shape. The non-figurate numbers were found using patterns observed among figurate numbers. We then analyzed and proved several mechanisms by which non-figurate numbers increase in succession. The procedural methods to find the non-figurate numbers were then converted into an explicit formula for the kth non-s-figurate number. Regarding usage of figurate numbers, the current applications of s-figurate numbers are used in computing probabilities, adding finite sums of objects, and some applications in iterations of computer programs. On the other hand, non-figurate numbers do not yet serve a purporse, but is rather much like the `monster’ function, a mathematical inquiry with no current application.
1. Definitions s-figurate number: The sequence of numbers corresponding to the number of points in a s-gon series like the ones below:
Figure 1: The 3-figurate numbers above are 1, 3, 6, 10, 15, 21, ....
Figure 2: The 5-figurate numbers above are 1, 5, 12, 22, .... non-s-figurate number: A sequence of all the positive integers not included in the sequence of s-figurate numbers. k: Will generally refer to the position of a non-figurate number s: Will generally refer to the number of sides on the regular planar polygon of which the figurate numbers are based off of. F(s,n): The function that produces the nth s-figurate number. F(s; k): The function that produces the kth non-s-figurate number. This was the function sought after. Binary Oscillator : The binary oscillator of some number is the ceiling function of the number minus the floor function of the same number. If the number is an integer, then this will output zero. Otherwise, this will output one. Gap: The subsets of numbers that are bounded by two consecutive s-figurate numbers. 70 | 2015-2016 | Volume 5
In this example, integers from 1 through 10 are shown with the triangular numbers crossed out. The result is the first 6 non-triangular numbers. The gaps here are the subsets [2], [4, 5], and [7, 8, 9]. Each subset of integers that is between the crossed out numbers is a gap. Endpoint of a Gap: The k value that corresponds to the non-s-figurate number that is exactly one less than an sfigurate number. Could also be known as the k value of the last integer in a gap. Adding the number of gaps, denoted as n, to k and adding 1 gave correct non-s-gurate values in our observations. A tentative equation we found was k + n + 1. An exception arose when the sum of the gap sizes was exactly k. In an exception like this, we did not have to add 1 to k +n. These observations formed the fundamental basis for the general formula by: 1) Developing an algebraic method to find the number of gaps before k for any s and k. 2) Forming the basis for the requirement of a binary oscillator to correct `perfect’ values of k by 1. This requires a method to determine n.
2. Formula for General s-Figurate Numbers Since the formula for the non-s-figurate numbers is based on the formula for the s-figurate numbers, there is a need to use the formula for these numbers. The general formula for the nth s-figurate number is:
2.1
Proof of General s-figurate Number Formula
Figure 3: The increase from the nth step of a triangular number to the n+1th is equal to n+1. From figure 1, it can be seen that the n + 1th iteration