Buggy python submission (verdict: Time Limit Exceeded):
C = -(10**9 + 7)


def main():
    from collections import defaultdict
    N = int(input())
    K = N//2
    A = [int(i) for i in input().split()]
    dp = defaultdict(lambda: C)
    for i in range(N+1):
        dp[(i, 0)] = 0
        if i != 0:
            dp[(i, 1)] = max(dp[(i-1, 1)], A[i-1])
    for i in range(2, N+1):
        for j in range((i+2)//2-1, (i+2)//2 + 1):
            if j < 0:
                continue
            cur = dp[(i, j)]
            for k in range(2, i+1-j):
                if 0 < i-k <= N:
                    cur = max(cur, dp[(i-k, j-1)] + A[i-1])
            dp[(i, j)] = cur
    print(dp[(N, K)])


if __name__ == '__main__':
    main()


Fixed submission:
C = -(10**15 + 7)


def main():
    from collections import defaultdict
    N = int(input())
    K = N//2
    A = [int(i) for i in input().split()]
    dp = defaultdict(lambda: C)
    for i in range(N+1):
        dp[i, 0] = 0
        if i != 0:
            dp[(i, 1)] = max(dp[i-1, 1], A[i-1])
    for i in range(2, N+1):
        for j in range(i//2-1, (i+1)//2 + 1):
            if j < 0:
                continue
            # A_i(1-index)でj個選ばなければいけない
            dp[i, j] = max(dp[i, j], dp[i-2, j-1] + A[i-1],
                           dp[i-3, j-1] + A[i-1], dp[i-4, j-1] + A[i-1])
    print(max(dp[N, K], dp[N-1, K], dp[N-2, K]))
    # for i in range(N+1):
    #     for j in range(K+1):
    #         print("{:>4}".format(dp[i, j] if dp[i, j] != C else "C"),
    #               end=" " if j != K else "\n")


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
C = -(10**9 + 7)


def main():
    from collections import defaultdict
    N = int(input())
    K = N//2
    A = [int(i) for i in input().split()]
    dp = defaultdict(lambda: C)
    for i in range(N+1):
        dp[(i, 0)] = 0
        if i != 0:
            dp[(i, 1)] = max(dp[(i-1, 1)], A[i-1])
    for i in range(2, N+1):
        for j in range(max(i//2 - 1, 0), i+1//2 + 1):
            dp[(i, j)] = max(dp[(i, j)], dp[(i-1, j)],
                             dp[(i-2, j-1)] + A[i-1],
                             dp[(i-3, j-1)] + A[i-1])
    print(dp[(N, K)])
    # print(*dp, sep="\n")


if __name__ == '__main__':
    main()


Fixed submission:
C = -(10**15 + 7)


def main():
    from collections import defaultdict
    N = int(input())
    K = N//2
    A = [int(i) for i in input().split()]
    dp = defaultdict(lambda: C)
    for i in range(N+1):
        dp[i, 0] = 0
        if i != 0:
            dp[(i, 1)] = max(dp[i-1, 1], A[i-1])
    for i in range(2, N+1):
        for j in range(i//2-1, (i+1)//2 + 1):
            if j < 0:
                continue
            # A_i(1-index)でj個選ばなければいけない
            dp[i, j] = max(dp[i, j], dp[i-2, j-1] + A[i-1],
                           dp[i-3, j-1] + A[i-1], dp[i-4, j-1] + A[i-1])
    print(max(dp[N, K], dp[N-1, K], dp[N-2, K]))
    # for i in range(N+1):
    #     for j in range(K+1):
    #         print("{:>4}".format(dp[i, j] if dp[i, j] != C else "C"),
    #               end=" " if j != K else "\n")


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
C = -(10**9 + 7)


def main():
    from collections import defaultdict
    N = int(input())
    K = N//2
    A = [int(i) for i in input().split()]
    dp = defaultdict(lambda: C)
    for i in range(N+1):
        dp[(i, 0)] = 0
        if i != 0:
            dp[(i, 1)] = max(dp[(i-1, 1)], A[i-1])
    for i in range(2, N+1):
        for j in range(max(i//2 - 1, 0), i+1//2 + 1):
            if dp[(i-1, j)] == C and dp[(i-2, j-1)] == C \
                    and (i < 3 or dp[(i-3, j-1)] == C):
                continue
            else:
                dp[(i, j)] = max(dp[(i, j)], dp[(i-1, j)],
                                 dp[(i-2, j-1)] + A[i-1],
                                 dp[(i-3, j-1)] + A[i-1])
    print(dp[(N, K)])
    # print(*dp, sep="\n")


if __name__ == '__main__':
    main()


Fixed submission:
C = -(10**15 + 7)


def main():
    from collections import defaultdict
    N = int(input())
    K = N//2
    A = [int(i) for i in input().split()]
    dp = defaultdict(lambda: C)
    for i in range(N+1):
        dp[i, 0] = 0
        if i != 0:
            dp[(i, 1)] = max(dp[i-1, 1], A[i-1])
    for i in range(2, N+1):
        for j in range(i//2-1, (i+1)//2 + 1):
            if j < 0:
                continue
            # A_i(1-index)でj個選ばなければいけない
            dp[i, j] = max(dp[i, j], dp[i-2, j-1] + A[i-1],
                           dp[i-3, j-1] + A[i-1], dp[i-4, j-1] + A[i-1])
    print(max(dp[N, K], dp[N-1, K], dp[N-2, K]))
    # for i in range(N+1):
    #     for j in range(K+1):
    #         print("{:>4}".format(dp[i, j] if dp[i, j] != C else "C"),
    #               end=" " if j != K else "\n")


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
C = -(10**9 + 7)


def main():
    N = int(input())
    K = N//2
    A = [int(i) for i in input().split()]
    dp = [[C]*(K+1) for _ in range(N+1)]
    for i in range(N+1):
        dp[i][0] = 0
        if i != 0:
            dp[i][1] = max(dp[i-1][1], A[i-1])
    for i in range(2, N+1):
        for j in range(2, K+1):
            if dp[i-1][j] == -1 and dp[i-2][j-1] == -1 \
                    and (i < 3 or dp[i-3][j-1] == -1):
                dp[i][j] = -1
            else:
                dp[i][j] = max(dp[i][j], dp[i-1][j],
                               dp[i-2][j-1] + A[i-1])
                if i > 2:
                    dp[i][j] = max(dp[i][j], dp[i-3][j-1] + A[i-1])
    print(dp[N][K])
    # print(*dp, sep="\n")


if __name__ == '__main__':
    main()


Fixed submission:
C = -(10**15 + 7)


def main():
    from collections import defaultdict
    N = int(input())
    K = N//2
    A = [int(i) for i in input().split()]
    dp = defaultdict(lambda: C)
    for i in range(N+1):
        dp[i, 0] = 0
        if i != 0:
            dp[(i, 1)] = max(dp[i-1, 1], A[i-1])
    for i in range(2, N+1):
        for j in range(i//2-1, (i+1)//2 + 1):
            if j < 0:
                continue
            # A_i(1-index)でj個選ばなければいけない
            dp[i, j] = max(dp[i, j], dp[i-2, j-1] + A[i-1],
                           dp[i-3, j-1] + A[i-1], dp[i-4, j-1] + A[i-1])
    print(max(dp[N, K], dp[N-1, K], dp[N-2, K]))
    # for i in range(N+1):
    #     for j in range(K+1):
    #         print("{:>4}".format(dp[i, j] if dp[i, j] != C else "C"),
    #               end=" " if j != K else "\n")


if __name__ == '__main__':
    main()