[Independant Discovery] The Pattern in the Sums of Digits of Perfect Squares

I remember sitting in algebra 1 class. In 8th grade, students hadn't been separated out much by ability and so my merciful teacher, Mrs. Parker, was kind enough to overlook the lack of attention paid her by some of us. Those who could afford to let our minds wonder were usually doing something quite nerdy, anyway. My friend, Brilliant Beautiful Beth, and I (take a moment to realize this was a different day and age) passed back and forth better and better (and oftentimes quite entertaining) designs for explosives. My boyfriend, Outstanding Oakland, sat at the back of BBB's row with a compass and a straight edge, sure he could find a method of forming a perfect triangle with only those tools. This is the part where a fraction of polymath readers think, "yeah, that was middle school," and another fraction think, "are these people from the same planet as me?" I assure you, everyone who has abilities in one thing certainly makes up for it by a lack somewhere else...Did anyone ever see me play basketball?

One day, I was playing with perfect squares. I wrote them down:
1, 4, 9, 16, 25, 36, 49, 64, 81...

Something popped out at me, for some reason:
One number was 9
One number was 36, and 3+6=9
One number was 81, and 8+1=9

I thought maybe this was a pattern, so I pulled back focus and looked at the short list again. I noticed that the last digits were:
1, 4, 9, 6, 5, 6, 9, 4, 1
It is a palindrome! (A word a had recently learned at quiz bowl, thank you Mrs. Hughes)
It turns out that I was identifying the diagonal of the Vedic Square.

I wondered what would happen if I continued it:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, (oh, a zero- that was exciting) 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 361…

The final digits went 1 4 9 6 5 6 9 4 1 0 1 4 9 6 5 6 9 4 1 0. I realized that 1 wasn’t the first perfect square- zero was, and so I grouped them: 0 (1496) 5 (6941) 0.

I went back to my original list of perfect squares, and in looking over it, half-heartedly added the digits in each number together (81:8+1=9, 64:6+4=10:1+0=0, etc), and another pattern began to emerge:

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 1 1 4 9 7 7 9 4 1 9

(1 4 9 7)-(7 9 4 1) 9 (1 4 9 7)-(7 9 4 1) 1 (1 4 9 7)-(7 9 4 1) 9

Beyond this point, I think displaying more lists of typed numbers will increasingly loose meaning, but you can get the idea by looking at the picture.

This was the first year my school had computers that went on the internet for us. (Yes, I'm that old) I didn't know how to talk about these things, let alone search for them. I showed this to my teacher, who in all fairness, didn't really pay attention. Turn about, of course, is fair play. Magnificent Mum smiled and told me that she loved me and that I was very smart. Otherwise, I set my little discovery aside. Once I got to college, though, it occurred to me to do a web search. Google (the new search engine that trumped yahoo/gopher/askjeeves) gave me two links when I searched it, both of which looked more like programming code than anything I knew how to understand.

Today, I decided to make a blog post, since I already had a scan of the thing that I had written neatly during some general requirement course in college. I re-googled, using the first full iteration of the pattern from the final column in the picture (1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1), and found only two websites that had it in any form: 

Some 2012 Forum Discussion
(btw- LOVE these people! In another life I hope to work for the Lone Gunmen)

A Mathematics Discussion Forum
(love these people, too, hence the totally unnecessary minor in applied mathematics)

Anyway, I figured Polymath Voyage is my bully pulpit, so I thought I'd share. :)