Let $a(n)$ be A079501 (i.e., number of compositions of the integer $n$ with strictly smallest part in the first position).

The sequence begins with

$$ 1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, 173, 275 $$

- Let

$$ b(n) = 1 + \sum\limits_{i=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\sum\limits_{j=0}^{\left\lfloor\frac{n-2i-1}{i+1}\right\rfloor}\binom{n - i(j+2) - 1}{j} $$

I conjecture that $$b(n)=a(n).$$

Here is the *PARI/GP* program to compute $b(n)$:

```
b(n) = 1 + sum(i = 1, n\2, sum(j=0, (n-2*i-1)\(i+1), binomial(n - i*(j+2) - 1, j)))
```

Is there a way to prove it?