This post explores some different ways of generating the famous Triangle Numbers with Python.

Before reading on, have a go a writing some Python code to print out the first 10 Triangle Numbers.

TRIANGLE NUMBERS USING A LIST

The first approach we will look at involves building up a list of Triangle Numbers by using a loop and the append list method.

n = 10
triangle_nums = []
tri_num = 0

for i in range(1, n + 1):
    tri_num += i
    triangle_nums.append(tri_num)

print(triangle_nums)

"""
Output: [1, 3, 6, 10, 15, 21, 28, 36, 45, 55]
"""

TRIANGLE NUMBERS USING THE FORMULA

There is a simple formula for calculating the nth Triangle Number, and it comes with a lovely and easy-to-understand proof, which you can learn about here.

The code is shown below. It uses a for loop and end="," inside the call to print(), to keep output on the same line, separated by commas.

for n in range(1, 11):
    print((n * (n + 1)) // 2, end=",")

"""
Output: 1,3,6,10,15,21,28,36,45,55,
"""

PARITY PATTERN IN TRIANGLE NUMBERS

Did you know that there is a pattern in the parity (oddness/evenness) of the triangle numbers?

You can see this from these results where T(n) is the nth Triangle Number, and P(T(n)) is its remainder on division by 2.

n       T(n)    P(T(n))
1        1       1
2        3       1
3        6       0
4        10      0
5        15      1
6        21      1
7        28      0
8        36      0
9        45      1
10       55      1
11       66      0
12       78      0
13       91      1
14       105     1

The code for producing the above results is:

print("n\tT(n)\tP(T(n))")  # \t is an escape character for tab
for n in range(1, 15):
    t = (n * (n + 1)) // 2
    p = t % 2
    print(n, "\t", t, "\t", p)

If you are into Maths, you may like to think about how this pattern arises. There are different ways to reason about it, but one involves the fact that all positive integers leave a remainder of either 0, 1, 2 or 3 on division by 4.

TRIANGLE NUMBERS USING A GENERATOR

Another approach, which is a great example for starting to learn about the Python keyword yield is shown below.

The yield keyword in Python used to exit from a function without disturbing the state of local variables, and when the function is called again, execution starts from the point where the code was left.

# Triangle numbers using a generator
def triangular_numbers(n):
    i, ti = 2, 1  # i is the counter, ti is the ith triangle number
    while i <= n + 1:
        yield ti
        ti += i
        i += 1


print(list(triangular_numbers(10)))

"""
Output: [1, 3, 6, 10, 15, 21, 28, 36, 45, 55]
"""

This post has shown some ways that we can use Python to generate Triangle Numbers. Each approach has its advantages and disadvantages, but they all produce the correct result. You may want to practice by trying them out for yourself now that you have seen them, and have a go at implementing one or more of them without looking back at this post.

Happy computing!

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