I'm interested in the limit as you take the square of the number given by the first $N$ digits of $\pi$ (ignoring the decimal point) & then add to it the first $M$ digits of $\pi$ (ignoring the decimal point again), where $M$ is the number of digits in the square of the first $N$ digits. Sorry if this sounds a bit confusing, but here are the examples:
N N^2 M N_M N^2 + N_M
1---------------3------------------------9---------------1------------------------3-----------------------------12
2--------------31---------------------961---------------3---------------------314--------------------------1275
3-------------314-----------------98596---------------5-------------------31415-----------------------130011
4------------3141--------------9865881---------------7----------------3141592 -------------------13007473
5-----------31415----------986902225---------------9-------------314159265 -----------------1301061490
6---------314159--------98695877281---------------11----------31415926535 --------------130111803816
7--------3141592---9869600294464---------------13--------3141592653589 ----------13011192948053
8------31415926-986960406437476--------------15------314159265358979 --------1301119671796455
What I'm interested in is, does $N^2 + N_M$ converge, and if so, to what value, and if not, how quickly divergent is it?
My best guess is that it converges, or is weakly divergent, but I'm extremely poorly-versed in limits, and know next to nothing about $\pi$'s properties or indeed formulas for it. But am curious nevertheless as to the result. Would appreciate any help. I am very far from my speciality here, which is integer arithmetic, algebra and diophantine equations. It's likely someone has investigated this before, but I don't really have a spare two hours or so to locate the answer using Google search, so would be grateful for any pointers.
Turning some comments into an answer (with a shift in notation for the number of digits involved), when you make an integer out of the first $n$ digits of $\pi$ (counting the $3$, which the comments below the OP do not), you have $\lfloor10^{n-1}\pi\rfloor$. Squaring this gives an integer with $2n-1$ digits. The integer formed by the first $2n-1$ digits of $\pi$ is $\lfloor10^{(2n-1)-1}\pi\rfloor$. So the OP's sequence (starting with $n=1$ is
$$\lfloor10^{n-1}\pi\rfloor^2+\lfloor10^{2n-2}\pi\rfloor$$
The $n$th term in the sequence has $2n$ digits. Roughly speaking, the first $n$ digits agree with the first $n$ digits of $\pi^2+\pi\approx13.011197054679\ldots$, while the final $n$ digits do not agree with the next $n$ digits of $\pi^2+\pi$. In particular, if you subtract the OP's sequence from $\lfloor10^{2n-2}(\pi^2+\pi)\rfloor$, you get the sequence
$$1,26,100,3724,56215,\ldots$$
This sequence is not (yet) in the OEIS. Aside from the $n$th term having (roughly?) $n$ digits, there's no apparent pattern in the numbers. I daresay there's no reason to expect one.