Triangular Square Numbers

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Triangular Square Numbers

Triangular Square Numbers

Aim

The aim of this investigation is to investigate numbers that are both square and triangular. Part of this investigation will be to try to find a formula to calculate the nth triangular square number.

Initial Analysis

Triangular numbers are numbers calculated using the equation:

Triangular square numbers must satisfy the above equation, and must also be square, so:

Obtaining Data

To investigate these numbers, some examples must be discovered. Finding these numbers by hand was tried, but it was found that triangular square numbers do not occur frequently in the set of natural numbers.

A computer program was written and ran to find these numbers quicker, and therefore be more efficient. This program is shown in Appendix A, and the output file in Appendix B. This program finds all the triangular square numbers between 1 and
1 x 10
16 – 1. It puts every number starting with 1 into the triangular formula, and tests if its square root is a whole number. If it is, the number produced by the triangular formula is triangular square.

This is the largest range that the programming language in which the program is written can handle with complete accuracy. The language is Gauss, which is actually designed for complex Economical calculations, but was chosen for this task because it handles numbers well. A more appropriate language to write this program in would be Maple, but a copy of the necessary program was not available.

Data Found

Note: The original triangular number is the number that is entered into the triangular formula to produce the triangular square number.

Analysing Data

The square roots of the triangular square numbers were analysed because they are smaller and so easier to handle. If a formula is found for the nth term in the sequence of square roots, this can then be squared to find the triangular square number.

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To find a formula, the differencing method was used, as seen in Appendix C, but a common difference was not found. This suggests that the formula is not a polynomial. The formula may be exponential. To check this, the ratios between successive terms should be analysed – or in other words each number in the sequence will be divided by the previous. This is shown in the following graph, for which the data can be found in Appendix D.

This shows that the ratio tends to a limit, which is a sign of exponential growth. Exponential growth is where ...

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