Expressing mathematically the number of real addition operation for a vector after dividing it

I assume I have the length of such vector $$y$$ is $$N$$. In the first time I divide that vector into two columns and then sum them point-wise summation. The second time, I divide the same vector $$y$$ into four columns and then sum them. Then, I divide $$y$$ into eight columns and sum them. I need to mathematically find the number of additions for that operation in function of $$N$$ and number of divisions. Assume that number of divisions is $$l$$.

Example

The vector $$y$$ with length $$N= 64$$, and $$l = 4$$.

$$l = 1$$, $$y$$ will be divided into two vectors of length $$32$$ and number of additions are 32;

$$l = 2$$, $$y$$ will be divided into four vectors of length $$16$$ and number of additions are 48;

$$l = 3$$, $$y$$ will be divided into eight vectors of length $$8$$ and number of additions are 56;

$$l = 4$$, $$y$$ will be divided into sixteen vectors of length $$4$$ and number of additions are 60;

So how can I express the total number of real addition mathematically in function of $$l$$ and $$N$$

• I’m voting to close this question because it does not seem to be about signal processing.
– MBaz
Jul 4, 2022 at 13:48

The number of additions is just the length of the individual vectors times their number minus $$1$$:
$$\textrm{number of additions}=N\cdot\frac{2^l-1}{2^l}$$
Of course we assume that $$N$$ is a power of $$2$$, and $$2^l\le N$$.