Congruence patterns in the series of triangular numbers

The Swiss occultist Oswald Wirth writes of what he calls "Theosophic addition and reduction." I don't know what it's got to do with Theosophy, but I found it mathematically interesting.

By "Theosophic addition," Wirth simply means the operation which yields the series of triangular numbers, where the nth triangular number is the sum of all integers from 0 to n, inclusive; in other words, the nth triangular number is equal to (n² + n) ÷ 2. The first several triangular numbers are: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666.

"Theosophic reduction" means adding up all the digits of a number, and then repeating the process as necessary until a one-digit number is arrived at. Mathematically, this amounts to finding the smallest positive integer to which it is congruent modulo 9.

Wirth took the first 21 triangular numbers (beginning with 1) and "Theosophically reduced" them, yielding this series: 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6. As you can see, the same series of 9 numbers (1, 3, 6, 1, 6, 3, 1, 9, 9) repeats itself; and if you keep going beyond 21 (which is where Wirth stopped because he was considering the numerology of the 21 Tarot trumps), it becomes apparent that it keeps repeating itself forever.

Wirth took this pattern as confirmation of the traditional (ultimately Pythagorean) numerological idea that the first 9 natural numbers are the building blocks of all the rest, and that that "the decad is a new monad." However, it seemed pretty obvious to me that there is nothing special about the number 9, and that the 9-based pattern Wirth found was almost certainly an artifact of the use of the decimal system in the "Theosophical reduction" -- such that "reduction" meant finding congruence modulo 9 -- and that other moduli would yield other patterns.

I also saw that Wirth had missed an interesting pattern in his integer series because he had started with 1 rather than 0, and because he was thinking in terms of "reduction" rather than congruence. (Nine is congruent to 0 modulo 9, but you can't arrive at that 0 by adding up digits in the "Theosophic" fashion.) If we start with the 0th triangular number (which is 0), and if we "reduce" multiples of 9 to 0 rather than to 9, the repeating series becomes (0, 1, 3, 6, 1, 6, 3, 1, 0) -- which is a palindrome!

I decided to check other moduli, starting with the easiest, which is 10. Any decimal number is congruent modulo 10 to its final digit, so look back at that list of triangular numbers and look at the final digits. The first thing you will notice is that certain final digits (2, 4, 7, 9) never occur at all. Look a little further, and you will see that there is a repeating pattern. It has a period of 20 (not 10, as we might have expected) -- and, sure enough, it is a palindrome: (0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0).

It is natural to jump from this to the induction that every modulus will yield a palindromic repeating pattern, and this turns out to be true for all the moduli I have looked at, as the table below shows. I have used centered alignment to highlight the palindromic nature of the repeating series.

From this sample, it appears that:

1. Any modulus, m, yields a repeating palindromic series of congruences for the series of triangular numbers.
2. If m is odd, the period of the repeating series is equal to m.
3. If m is even, the period is equal to 2m, and each of the two numbers at the center of the palindrome is equal to m/2.

Now, I'm roughly 100% sure that I'm not the first person to have noticed these patterns, and that someone else has already mathematically proven what I have only induced. So, to those of my readers who have had a proper mathematical education -- no spoilers, please! I want to try to figure this out for myself.