# Triangle numbers

## 0. Preliminary

**Exercise**

Write a function to find the triangle sum of any positive integer. The triangle sum of \(n\) is the arithmetic sequence \(1 + 2 + 3 + \dots + (n-1) + n\).

**Additional** information in this module

VBA skills:

- loops - the For...Next construct
- writing function procedures

What is meant by the "triangle sum"?

"A triangular number or triangle number counts objects arranged in an equilateral triangle" (Source: Wikipedia 2017)

Let \(n\) represent the number of dots *on one side of the triangle*, then the triangle number \(T_n\) has the formula $$\begin{equation}T_n=\sum_{k=1}^n = 1+2+3+\dots +(n-1)+n\end{equation}$$ see figure 1 were the dots *on one side of the triangle* can be envisaged as rows of dots of the same colour.

The sequence continues in figure 2.

## 1. Suggested solution

**Code 1:**Macro

`Triangle`

write the data range for triangle n
Function Triangle(n As Long) As Long ' This function calculates the triangular sum of a positive number Dim x As Long, y As Long y = 0 For x = 1 To n y = y + x Next x Triangle = y End Function

' Version 2 with error trapping for non positive n Function Tri2(n As Integer) As Variant Dim x As Long, y As Long If n < 1 Then Tri2 = CVErr(xlErrValue) Else y = 0 For x = 1 To n y = y + x Next x Tri2 = y End If End Function

## 2. Creating the triangle charts in Excel

A minimum of three data points \(n => 2\) are required to create a triangle X Y chart.

### 2.1 Equilateral Triangle

The chart in figure 3 is an Excel X Y scatter chart with 10 rows. You can click the chart to view the data set.

### Create the data set

The data points for the chart in figure 1 were written to the worksheet with the **TriangleData** macro in code 2.

**Code 2:**Macro

`TriangleData`

write the data range for triangle n (min 2)
Sub TriangleData() ' # ' # # ' # # # ' # # # # ' interface connectors Const n As Integer = 10 Const AnchorRef As String = "B4" Dim x As Integer, y As Integer Dim Anchor As Range Dim Mid As Integer Dim Max As Integer Max = n * 2 - 1 ' x axis 1 to Max (an odd number) Set Anchor = Range(AnchorRef) ' write x vector For x = 1 To Max Anchor(x, 1).Value = x Next x Mid = (Max + 1) / 2 ' mid point of x vector ' write y vectors For y = 0 To n - 1 For x = Mid - y To Mid + y Step 2 Anchor(x, y + 2).Value = y + 1 Next x Next y ' select x-y data set (reverse order in chart axis) Range(AnchorRef, Cells(x + 1, y + 2)).Select End Sub

### Add the X Y (scatter) chart

To add the X Y chart:

- Select the data if not already selected by line 35 of code 2
- On the ribbon, select - the new chart will appear to be inverted with the point of the triangle at the base
- Right click the chart to display the (
**Chart Area**) context menu and select to display the**Format Chart Area pane**, then select Vertical (Value) Axis item - In the
**Axis Options**section, tick - Adjust other format settings as appropriate

### 2.2 Lower Triangle

This version of the triangle is relevant to aspects of finance such as the portfolio variance-covariance matrix.

### Create the data set

The data points for the chart in figure 2 were written to the worksheet with the **TriangleData2** macro in code 3.

**Code 3:**Macro

`TriangleData2`

write the data range for triangle n (min 2)
Sub TriangleData2() ' # ' # # ' # # # ' # # # # ' interface connectors Const n As Integer = 10 Const AnchorRef As String = "B24" Dim x As Integer, y As Integer, z As Integer Dim Anchor As Range Dim Max As Integer Max = n * 2 - 1 ' x axis 1 to Max (an odd number) Set Anchor = Range(AnchorRef) ' write x vector For x = 1 To Max Anchor(x, 1).Value = x Next x ' write y vectors For x = 1 To Max Step 2 z = z + 1 For y = z To n Anchor(x, y + 1).Value = y Next y Next x ' select x-y data set Range(AnchorRef, Cells(x + 1, y + 2)).Select End Sub

To add the X Y chart, follow the steps in section 2.1.

## 3. Finance aspects

Equation 1 can be written as \(n(n+1)/2\) as shown in equation 2 $$\begin{equation}T_n=\sum_{k=1}^n=1+2+3+\dots +(n-1)+n=\frac{n(n+1)}{2}=n(n+1)/2\end{equation}$$ For \(n = 10\), \(T_{10} = (10 \times 11)/2 = 55\)

The lower triangle in figure 2 is equivalent to a lower triangle of a variance matrix. For a 10 asset portfolio there are 55 variance-covariance items. The number of portfolio covariance terms, items off the main diagonal is \(T_n-n=55-10=45\) $$\begin{equation}T_n-n =\frac{n(n+1)}{2}-n=\frac{n^2+n}{2}-\frac{2n}{2}=\frac{n^2-2n+n}{2}=\frac{n(n-1)}{2}=n(n-1)/2\end{equation}$$ For \(n = 10\), \(T_{10}-n =(10 \times 11)/2-10=55-10=45\) covariance items (off diagonal) and \(10\) variance items (on diagonal) - see figure 5.

Other applications in finance include, Pascal's triangle, and binomial trees.

**References**- Excel at Finance: Convert a lower triangular table to full matrix Accessed: 4 November 2017
- Wikipedia: Triangular number Accessed: 4 November 2017

**Development platform:**Excel 2016 (64 bit) Office 365 ProPlus on Windows 10**Revised:**Wednesday 4th of July 2018 - 09:31 AM, [Australian Eastern Time (AET)]