What is the history behind triangular sequences?

Answer 1

Pythagoras studied these numbers thousands of years ago so the history is very long and detailed. Since the nth triangular number is the sum of the numbers before it including that numbers, we can write the sequence of triangular numbers and denote the nth term in the sequence by an and the sum of the first n numbers is 1/2(n)(n+1). We can also write it using the notation (n+1)C2 which means pick 2 from n+1. So if we have n=3 it is 4C2 which is 4x3/2 or 6
For example a3 =1/2(3x4)=6. This can also be found be adding 1+2+3=6

Gauss noted that every positive integer can be written as the sum of at most 3 triangular numbers.

There numbers are also diagonals in pascals triangle.

Have a look at the nice video which gives more info and history.