# Big $O$ approximation for $T(n)=T(n-i)+T(n-(\frac{n}{m}-i))$

I have the following complexity equation: $$T(n)=T(n-i)+T(n-(\frac{n}{m}-i))$$ with the base case $$T(m)=1$$.

Is it possible to calculate a big $$O$$ approximation for such equation? What is the right method to do so?

Edit: I actually need to calculate this equation for any $$1\leq{i}<\frac{n}{m}$$ so if it is easier to consider $$i$$ as a constant, such a solution is also helpful.

Edit 2: I think the equation is also missing floor value of $$\frac{n}{m}$$ so it will be coherent

• I think $i$ should be between $1$ and $\frac{n}{m}$. Or at least $n-m$ should be less than $\frac{n}{m}$. Right? – OmG Jun 24 at 12:56
• yes, thanks for the correction I'll edit the main post – Ofir Gordon Jun 24 at 13:23
• Thank you. One more thing. Is $m$ constant? As you have mentioned that $T(m) = 1$. – OmG Jun 24 at 13:40
• Is it not required that $i<{n\over m}$ for all n, so $n-{n\over m} + i<n$ would hold for all n? It would directly imply that $i=1$. I suppose you have meant something else - maybe $i$ depends on $n$? then write it at $i_n$ for clarification. – nir shahar Jun 24 at 14:13
• @OmG yes let's consider $m$ as a constant. Does that mean the result wouldn't be dependent on $m$? – Ofir Gordon Jun 24 at 16:56