How to solve the grasshopper problem correctly using dynamic programming?

Question

Python

Hello! Wrote code to solve the following problem:

Grasshopper
Grasshopper jumps over flowers located on a straight line. Each flower contains pollen. At the beginning of the path, the grasshopper is on the ground.

Grasshopper can

• jump to the next flower
  
• jump over one flower
  

How can a grasshopper get from the ground to the last flower with less pollen.

Input data format
Two lines. In the first one - the number of flowers, N. In the second array of length N - the amount of pollen (from 1 to 100) on each flower.

Output data format
Two lines. In the first, the minimum amount of pollen that a grasshopper will get dirty with. In the second - the path of the grasshopper. The path represents the numbers of flowers that the grasshopper will visit. Flower numbering from scratch.

Examples
Input
3
1 3 1
Output
2
0 2

Input
4
2 1 5 1
Output
2
1 3

The problem with my code is that it fails some tests that are hidden from me. I checked the code on the notebook manually, everything seems to be working. Please explain what is the problem. Thanks in advance!

def minimum(F: list):
    pollen_array = []
    if len(F) < 3:
        pollen_array.append(0)
        return F[1], pollen_array
    elif len(F) == 3:
        pollen_array.append(F.index(min(F[1], F[2])) - 1)
        return min(F[1], F[2]), pollen_array
    else:
        pollen = 0
        i = 0
        while i < len(F) - 1:
            if i < len(F) - 2:
                pollen += min(F[i + 1], F[i + 2])
                i = F.index((min(F[i + 1], F[i + 2])), i + 1) if F[i + 1] != F[i+2] else i + 2
                pollen_array.append(i - 1)
            elif i == len(F) - 2:
                pollen += F[i + 1]
                pollen_array.append(i)
                break
    return pollen, pollen_array


n = int(input())
F = list(map(int, input().split()))
F.insert(0, 0)

a, b = minimum(F)
print(a)
print(*b)

Answer 1

You understood that this is dynamic programming. Describe what kind of DP function you have, give the recalculation formula, base and where is the answer.