There cannot be 0 in the split, otherwise there are infinite split methods: 4 = 4+0 = 4+0+0 = ...
Directly think that 4 = 4 is also a kind of split.
Let p(n) represent the total number of n splits, which supplements the definition of arbitrary integer n with p(0) = 1 and p(n) = 0.
P(n) doesn't have a closed general formula, so I don't think so.
It is easy to get a formal power series (generating function) with p(n) as the coefficient.
∑{ n≥0 } p(n)x^n =п{ n≥ 1 }( 1+x^n+x^(2n)+x^(3n)+...)=п{ n≥ 1 } 1/( 1-x^n).
Combined with Euler's pentagonal number theorem, a recursive formula of p(n) can be obtained:
P(n) = ∑{k is a non-zero integer} (-1) (k-1) p (n-k (3k-1)/2)
=∑{ k≥ 1 }(- 1)^(k- 1)p(n-k(3k- 1)/2)+∑{ k≥ 1 }(- 1)^(k- 1)p(n-k(3k+ 1)/2)。
Note that when k is a non-zero integer, k (3k-1)/2 >; 0, in addition, there is only a finite integer k that makes k (3k-1)/2 ≤ n. 。
Therefore, only integers less than n appear in sum, and only finite items are not zero.
In addition, p(n) has a gradual formula (when n→∞, the ratio of both sides tends to 1):
e^(π√(2n/3))/(4n√3).