Sometimes I just like to play with numbers:
12 + 9*12 = 10
82 + 9*22 = 102
262 + 9*62 = 103
282 + 9*322 = 104
Which goes out to infinity, where I used a simple rule.
If: u2 + 9v2 = 10a
Then: (u - 9v)2 + 9(u + v)2 = 10a+1
Which is just using my BQD Iterator. And notice that u and v can be positive or negative, while I like to show positive as is easier and looks prettier, which allows some selectivity, which I used behind the scenes to get my series above. Some choices made it more boring.
Like yeah so if you use a = 4, u = 28, and v = -32, then next is:
3162 + 9*42 = 105
So yeah, every power of 10 can be written as a sum of two squares.
And was fascinated a couple of years ago that the general result is, for an integer n equal to 0 or higher, and an integer m equal to 3 or higher:
x2 + (m-1)y2 = mn+1
I have m raised to n+1 so that n is a count of iterations. And if m-1 is a square then every power of m can be shown as the sum of two squares.
And talk it all out in this post. So I just used m = 10 above.