I saw this too, and it would have been much easier to use. When I noticed the pattern (1+2=3, 4+5+6=7+8...), the perfect squares stuck out and then when I came up with the recursive function I was looking at it more from a point of view of the squares than I was from the pov of a sequense...
T(n) = (T( (n^(1/2) -1 )^2 )/3 + n) * 3
T(0) = 0
T(1) = 3
T(4) = 15
T(3^2) = ( T(2^2) )/3 + 3^2) * 3 = (15/3 + 9) * 3 = 42
I think the problem was that I knew what I was thinking but then I didn't clearly state it. I was intending that T(n-1) would be the next lowest perfect square...
I think that would make it simpler too. Before I figured out the recursive function, I was using excel with the formula n + (n+1) + ... + (n+n^(1/2)), that was a pain because you can't just copy and paste...
I miss stated it, it's: T(0) = 0, T(n) = (T( (n^(1/2) -1 )^2 )/3 + n) * 3, when n^(1/2) is an integer.
so for n = 4, its T(4) = ( T(1)/3 + 4 )*3 = (3/3 + 4)*3 = 15
It can also be n + (n + 1) + ... + (n + n^(1/2)) = (n + n^(1/2) + 1) + ... + ( n + 2*n^(1/2) )
That works too, the formula I was using is T(0) = 0, T(n) = (T(n-1))/3 + n) * 3. The interesting part was that if you add consecutive numbers, they add to the next set of consecutive numbers, starting with a perfect square and going to the number before the next perfect square. Like 1+2 = 3...
0, 3, 15, 42, 90, ...
What number comes next?
There's something interesting about these numbers (at least to me). <- that's not a clue its just that this isn't just some random series, there's more to it that is mathematical, not personal or a bus route...
Feynman did some experiments with this and talks about it in one of his 'curious character' books- I can't remember which one. Anyway, he smoked pot and got in a tank that was able to remove his senses by removing external stimuli (I think the tank was designed, not by Feynman, to try to mimic...