Buggy python submission (verdict: Time Limit Exceeded):
from collections import deque, Counter, defaultdict
from itertools import chain, combinations
import json
# import numpy as np
import bisect
import sys
import math
import bisect
from functools import lru_cache

sys.setrecursionlimit(10 ** 8)

M = 10 ** 9 + 7
INF = 10 ** 17


def main():
    N = int(input())
    A = [int(a) for a in input().split()]

    @lru_cache(None)
    def f(i, j):
        if i < 0 or j < 0:
            return -INF
        if i == 0:
            return -INF
        if j == 0:
            return 0
        if j == 1:
            return max(A[:i])
        return max(A[i - 1] + f(i - 2, j - 1), f(i - 1, j))

    print(f(N, math.floor(N / 2)))


if __name__ == "__main__":
    main()


Fixed submission:
from collections import deque, Counter, defaultdict
from itertools import chain, combinations
import json
# import numpy as np
import bisect
import sys
import math
import bisect
from functools import lru_cache

sys.setrecursionlimit(10 ** 8)

M = 10 ** 9 + 7
INF = 10 ** 17


def main():
    N = int(input())
    A = [int(a) for a in input().split()]

    @lru_cache(None)
    def f(i, j):
        if i < 0 or j < 0:
            return -INF
        if i == 0:
            return -INF
        if abs(j - i // 2 - 1) >= 2:
            return -INF
        if j == 0:
            return 0
        if j == 1:
            return max(A[:i])
        return max(A[i - 1] + f(i - 2, j - 1), f(i - 1, j))

    print(f(N, math.floor(N / 2)))


if __name__ == "__main__":
    main()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import deque, Counter, defaultdict
from itertools import chain, combinations
import json
# import numpy as np
import bisect
import sys
import math
import bisect
from functools import lru_cache

sys.setrecursionlimit(10 ** 8)

M = 10 ** 9 + 7
INF = 10 ** 17


def main():
    N = int(input())
    A = [int(a) for a in input().split()]

    @lru_cache()
    def f(i, j):
        if i < 0 or j < 0:
            return -INF
        if i == 0:
            return -INF
        if j == 0:
            return 0
        if j == 1:
            return max(A[:i])
        return max(A[i - 1] + f(i - 2, j - 1), f(i - 1, j))

    print(f(N, math.floor(N / 2)))


if __name__ == "__main__":
    main()


Fixed submission:
from collections import deque, Counter, defaultdict
from itertools import chain, combinations
import json
# import numpy as np
import bisect
import sys
import math
import bisect
from functools import lru_cache

sys.setrecursionlimit(10 ** 8)

M = 10 ** 9 + 7
INF = 10 ** 17


def main():
    N = int(input())
    A = [int(a) for a in input().split()]

    @lru_cache(None)
    def f(i, j):
        if i < 0 or j < 0:
            return -INF
        if i == 0:
            return -INF
        if abs(j - i // 2 - 1) >= 2:
            return -INF
        if j == 0:
            return 0
        if j == 1:
            return max(A[:i])
        return max(A[i - 1] + f(i - 2, j - 1), f(i - 1, j))

    print(f(N, math.floor(N / 2)))


if __name__ == "__main__":
    main()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import deque, Counter
from itertools import chain, combinations
import json
# import numpy as np
import bisect
import sys
import math
import bisect

sys.setrecursionlimit(10 ** 8)

M = 10 ** 9 + 7


def main():
    N = int(input())
    A = [int(a) for a in input().split()]

    DP = [[0] * (N + 1) for _ in range(N + 1)]
    DP[1][1] = A[0]

    for i in range(2, N + 1):
        a = math.floor((i - 1) / 2)
        b = math.floor((i + 1) / 2)
        for j in range(a, b + 1):
            DP[i][j] = max(DP[i - 1][j], DP[i - 2][j - 1] + A[i - 1])
    print(int(DP[N][math.floor(N / 2)]))


if __name__ == "__main__":
    main()


Fixed submission:
from collections import deque, Counter, defaultdict
from itertools import chain, combinations
import json
# import numpy as np
import bisect
import sys
import math
import bisect
from functools import lru_cache

sys.setrecursionlimit(10 ** 8)

M = 10 ** 9 + 7
INF = 10 ** 17


def main():
    N = int(input())
    A = [int(a) for a in input().split()]

    @lru_cache(None)
    def f(i, j):
        if i < 0 or j < 0:
            return -INF
        if i == 0:
            return -INF
        if abs(j - i // 2 - 1) >= 2:
            return -INF
        if j == 0:
            return 0
        if j == 1:
            return max(A[:i])
        return max(A[i - 1] + f(i - 2, j - 1), f(i - 1, j))

    print(f(N, math.floor(N / 2)))


if __name__ == "__main__":
    main()

Buggy python submission (verdict: Runtime Error):
from collections import deque, Counter
from itertools import chain, combinations
import json
import numpy as np
import bisect
import sys
import math
import bisect

sys.setrecursionlimit(10 ** 8)

M = 10 ** 9 + 7


def main():
    N = int(input())
    A = [int(a) for a in input().split()]

    DP = np.zeros((N + 1, N + 1))
    DP[1][1] = A[0]

    for i in range(2, N + 1):
        a = math.floor((i - 1) / 2)
        b = math.floor((i + 1) / 2)
        for j in range(a, b + 1):
            DP[i][j] = max(DP[i - 1][j], DP[i - 2][j - 1] + A[i - 1])
    print(int(DP[N][math.floor(N / 2)]))


if __name__ == "__main__":
    main()


Fixed submission:
from collections import deque, Counter, defaultdict
from itertools import chain, combinations
import json
# import numpy as np
import bisect
import sys
import math
import bisect
from functools import lru_cache

sys.setrecursionlimit(10 ** 8)

M = 10 ** 9 + 7
INF = 10 ** 17


def main():
    N = int(input())
    A = [int(a) for a in input().split()]

    @lru_cache(None)
    def f(i, j):
        if i < 0 or j < 0:
            return -INF
        if i == 0:
            return -INF
        if abs(j - i // 2 - 1) >= 2:
            return -INF
        if j == 0:
            return 0
        if j == 1:
            return max(A[:i])
        return max(A[i - 1] + f(i - 2, j - 1), f(i - 1, j))

    print(f(N, math.floor(N / 2)))


if __name__ == "__main__":
    main()