Buggy python submission (verdict: Time Limit Exceeded):
import sys
import math
from collections import defaultdict
from bisect import bisect_left, bisect_right

sys.setrecursionlimit(10**7)
def input():
    return sys.stdin.readline()[:-1]

mod = 10**9 + 7

def I(): return int(input())
def LI(): return list(map(int, input().split()))
def LIR(row,col):
    if row <= 0:
        return [[] for _ in range(col)]
    elif col == 1:
        return [I() for _ in range(row)]
    else:
        read_all = [LI() for _ in range(row)]
        return map(list, zip(*read_all))

#################

def calc(i,j):
    d = defaultdict(lambda:defaultdict(int))
    ret = -float('inf')
    if j==0:
        return 0
    if (N-i+1)//2 < j:
        return ret
    else:
        if i <= N-1:
            if d[i+2][j-1]:
                ret = max(ret,d[i+2][j-1])
            else:
                ret = max(ret,A[i]+calc(i+2,j-1))
        if i+1 <= N-1:
            if d[i+3][j-1]:
                ret = max(ret,d[i+3][j-1])
            else:
                ret = max(ret,A[i+1]+calc(i+3,j-1))
        d[i][j] = ret
        return ret


N = I()
A = LI()


if N%2 == 0:
    print(calc(0,N//2))
else:
    a1 = calc(0,N//2)
    n = 0
    for i in range(2,N,2):
        n += A[i]
    print(max(a1,n))


Fixed submission:
import sys
import math
from collections import defaultdict
from bisect import bisect_left, bisect_right

sys.setrecursionlimit(10**7)
def input():
    return sys.stdin.readline()[:-1]

mod = 10**9 + 7

def I(): return int(input())
def LI(): return list(map(int, input().split()))
def LIR(row,col):
    if row <= 0:
        return [[] for _ in range(col)]
    elif col == 1:
        return [I() for _ in range(row)]
    else:
        read_all = [LI() for _ in range(row)]
        return map(list, zip(*read_all))

#################

N = I()
A = LI()


if N%2 == 0:
    d = defaultdict(lambda:defaultdict(lambda:-float('inf')))
    d[0][1] = A[0]
    d[1][1] = max(A[0],A[1])
    for i in range(2,N):
        if i%2 == 0:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
        else:
            for j in [(i+1)//2]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
    print(d[N-1][N//2])
else:
    d = defaultdict(lambda:defaultdict(lambda:-float('inf')))
    d[0][1] = A[0]
    d[1][1] = max(A[0],A[1])
    for i in range(2,N):
        if i%2 == 0:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
        else:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
    a1 = d[N-1][N//2]
    n = 0
    for i in range(2,N,2):
        n += A[i]
    print(max(n,a1))

Buggy python submission (verdict: Time Limit Exceeded):
import sys
import math
from collections import defaultdict
from bisect import bisect_left, bisect_right

sys.setrecursionlimit(10**7)
def input():
    return sys.stdin.readline()[:-1]

mod = 10**9 + 7

def I(): return int(input())
def LI(): return list(map(int, input().split()))
def LIR(row,col):
    if row <= 0:
        return [[] for _ in range(col)]
    elif col == 1:
        return [I() for _ in range(row)]
    else:
        read_all = [LI() for _ in range(row)]
        return map(list, zip(*read_all))

#################

def calc(i,j):
    d = defaultdict(lambda:defaultdict(int))
    ret = -float('inf')
    if j==0:
        return 0
    if (N-i+1)//2 < j:
        return ret
    else:
        if i <= N-1:
            if d[i+2][j-1]:
                ret = max(ret,d[i+2][j-1])
            else:
                ret = max(ret,A[i]+calc(i+2,j-1))
        if i+1 <= N-1:
            if d[i+3][j-1]:
                ret = max(ret,d[i+3][j-1])
            else:
                ret = max(ret,A[i+1]+calc(i+3,j-1))
        d[i][j] = ret
        return ret


N = I()
A = LI()


if N%2 == 0:
    print(calc(0,N//2))
else:
    a1 = calc(0,N//2)
    n = 0
    for i in range(2,N,2):
        n += A[i]
    print(max(a1,n))

Fixed submission:
import sys
import math
from collections import defaultdict
from bisect import bisect_left, bisect_right

sys.setrecursionlimit(10**7)
def input():
    return sys.stdin.readline()[:-1]

mod = 10**9 + 7

def I(): return int(input())
def LI(): return list(map(int, input().split()))
def LIR(row,col):
    if row <= 0:
        return [[] for _ in range(col)]
    elif col == 1:
        return [I() for _ in range(row)]
    else:
        read_all = [LI() for _ in range(row)]
        return map(list, zip(*read_all))

#################

N = I()
A = LI()


if N%2 == 0:
    d = defaultdict(lambda:defaultdict(lambda:-float('inf')))
    d[0][1] = A[0]
    d[1][1] = max(A[0],A[1])
    for i in range(2,N):
        if i%2 == 0:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
        else:
            for j in [(i+1)//2]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
    print(d[N-1][N//2])
else:
    d = defaultdict(lambda:defaultdict(lambda:-float('inf')))
    d[0][1] = A[0]
    d[1][1] = max(A[0],A[1])
    for i in range(2,N):
        if i%2 == 0:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
        else:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
    a1 = d[N-1][N//2]
    n = 0
    for i in range(2,N,2):
        n += A[i]
    print(max(n,a1))

Buggy python submission (verdict: Time Limit Exceeded):
import sys
import math
from collections import defaultdict
from bisect import bisect_left, bisect_right

sys.setrecursionlimit(10**7)
def input():
    return sys.stdin.readline()[:-1]

mod = 10**9 + 7

def I(): return int(input())
def LI(): return list(map(int, input().split()))
def LIR(row,col):
    if row <= 0:
        return [[] for _ in range(col)]
    elif col == 1:
        return [I() for _ in range(row)]
    else:
        read_all = [LI() for _ in range(row)]
        return map(list, zip(*read_all))

#################

def calc(i,j):
    ret = -float('inf')
    if j==0:
        return 0
    if (N-i+1)//2 < j:
        return ret
    else:
        if i <= N-1:
            ret = max(ret,A[i]+calc(i+2,j-1))
        if i+1 <= N-1:
            ret = max(ret,A[i+1]+calc(i+3,j-1))
        return ret


N = I()
A = LI()


if N%2 == 0:
    print(calc(0,N//2))
else:
    a1 = calc(0,N//2)
    n = 0
    for i in range(2,N,2):
        n += A[i]
    print(max(a1,n))

Fixed submission:
import sys
import math
from collections import defaultdict
from bisect import bisect_left, bisect_right

sys.setrecursionlimit(10**7)
def input():
    return sys.stdin.readline()[:-1]

mod = 10**9 + 7

def I(): return int(input())
def LI(): return list(map(int, input().split()))
def LIR(row,col):
    if row <= 0:
        return [[] for _ in range(col)]
    elif col == 1:
        return [I() for _ in range(row)]
    else:
        read_all = [LI() for _ in range(row)]
        return map(list, zip(*read_all))

#################

N = I()
A = LI()


if N%2 == 0:
    d = defaultdict(lambda:defaultdict(lambda:-float('inf')))
    d[0][1] = A[0]
    d[1][1] = max(A[0],A[1])
    for i in range(2,N):
        if i%2 == 0:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
        else:
            for j in [(i+1)//2]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
    print(d[N-1][N//2])
else:
    d = defaultdict(lambda:defaultdict(lambda:-float('inf')))
    d[0][1] = A[0]
    d[1][1] = max(A[0],A[1])
    for i in range(2,N):
        if i%2 == 0:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
        else:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
    a1 = d[N-1][N//2]
    n = 0
    for i in range(2,N,2):
        n += A[i]
    print(max(n,a1))

Buggy python submission (verdict: Time Limit Exceeded):
import sys
import math
from collections import defaultdict
from bisect import bisect_left, bisect_right

sys.setrecursionlimit(10**7)
def input():
    return sys.stdin.readline()[:-1]

mod = 10**9 + 7

def I(): return int(input())
def LI(): return list(map(int, input().split()))
def LIR(row,col):
    if row <= 0:
        return [[] for _ in range(col)]
    elif col == 1:
        return [I() for _ in range(row)]
    else:
        read_all = [LI() for _ in range(row)]
        return map(list, zip(*read_all))

#################

def calc(i,j):
    ret = -float('inf')
    if j==0:
        return 0
    if (N-i+1)//2 < j:
        return ret
    else:
        if i <= N-1:
            ret = max(ret,A[i]+calc(i+2,j-1))
        if i+1 <= N-1:
            ret = max(ret,A[i+1]+calc(i+3,j-1))
        return max(ret,calc(i+2,j))


N = I()
A = LI()

s = sum(A)
cums = [A[0]]
for i in range(1,N):
    cums.append(cums[-1]+A[i])

if N%2 == 0:
    print(calc(0,N//2))
else:
    print(calc(0,N//2))

Fixed submission:
import sys
import math
from collections import defaultdict
from bisect import bisect_left, bisect_right

sys.setrecursionlimit(10**7)
def input():
    return sys.stdin.readline()[:-1]

mod = 10**9 + 7

def I(): return int(input())
def LI(): return list(map(int, input().split()))
def LIR(row,col):
    if row <= 0:
        return [[] for _ in range(col)]
    elif col == 1:
        return [I() for _ in range(row)]
    else:
        read_all = [LI() for _ in range(row)]
        return map(list, zip(*read_all))

#################

N = I()
A = LI()


if N%2 == 0:
    d = defaultdict(lambda:defaultdict(lambda:-float('inf')))
    d[0][1] = A[0]
    d[1][1] = max(A[0],A[1])
    for i in range(2,N):
        if i%2 == 0:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
        else:
            for j in [(i+1)//2]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
    print(d[N-1][N//2])
else:
    d = defaultdict(lambda:defaultdict(lambda:-float('inf')))
    d[0][1] = A[0]
    d[1][1] = max(A[0],A[1])
    for i in range(2,N):
        if i%2 == 0:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
        else:
            for j in [i//2,i//2+1]:
                d[i][j] = max(d[i-2][j-1]+A[i],d[i-1][j])
    a1 = d[N-1][N//2]
    n = 0
    for i in range(2,N,2):
        n += A[i]
    print(max(n,a1))