# Task

Christmas is coming. In the [previous kata](https://www.codewars.com/kata/5a405ba4e1ce0e1d7800012e), we build a custom Christmas tree with the specified characters and the specified height.

Now, we are interested in the center of the Christmas tree. 

Please imagine that we build a Christmas tree with `chars = "abc" and n = Infinity`, we got:
```
          a
         b c
        a b c
       a b c a
      b c a b c
     a b c a b c
    a b c a b c a
   b c a b c a b c
  a b c a b c a b a
 b c a b c a b a b c
a b c a b a b c a b c
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
          |
          |
          .
          .
```
If we keep only the center part of leaves, we will got:
```
          a
            
          b  
              
          a    
                
          a        
                    
          b        
                    
          a          
          .
          
          .
          
          .
```
As you can see, it's a infinite string, but it has a repeat substring "aba"(spaces will be removed). If we join them together, it looks like:`"abaabaabaabaaba......"`.

So, your task is to find the repeat substring of the center part of leaves.

# Inputs:

- `chars`: the specified characters. In this kata, they always be lowercase letters.

# Output:

- The repeat substring that satisfy the above conditions.

Still not understand the task? Look at the following example ;-)

# Examples


For `chars = "abc"`,the output should be `"aba"`
```
center leaves sequence: "abaabaabaabaabaaba....."
```
For `chars = "abab"`,the output should be `a`
```
center leaves sequence: "aaaaaaaaaa....."
```
For `chars = "abcde"`,the output should be `aecea`
```
center leaves sequence: "aeceaaeceaaecea....."
```
For `chars = "aaaaaaaaaaaaaa"`,the output should be `a`
```
center leaves sequence: "aaaaaaaaaaaaaa....."
```
For `chars = "abaabaaab"`,the output should be `aba`
```
center leaves sequence: "abaabaabaaba....."
```

Difficulty: EASY
Tags: String algorithms

Fundamentals

Mathematics

Ad-hoc

Starter code:
def center_of(chars):
	

Solutions:
def center_of(chars):
	if not chars:
		return ''
	total = 0
	res = []
	for i in range(1, len(chars) * 2 + 1):
		if i % 2 == 1:
			res.append((i + 1) // 2 + total)
			res[-1] = chars[(res[-1] - 1) % len(chars)]
		total += i
	res = ''.join(res)
	for i in range(len(res) // 2 + 1):
		if len(res) % len(res[:i + 1]) != 0:
			continue
		if res[:i + 1] * (len(res) // len(res[:i + 1])) == res:
			return res[:i + 1]
	return res


from itertools import count

def center_of(s):
	if not s:
		return ''
	(seq, le, i) = ('', len(s), 0)
	for k in count(1):
		seq += s[i % le]
		i += 4 * k
		n = len(seq) / 3
		if n.is_integer():
			n = int(n)
			if seq[:n] == seq[n:n * 2] == seq[n * 2:] and i >= le:
				return seq[:n]


from collections import deque
from itertools import count, islice

def iter_center(s):
	q = deque(s)
	for i in count():
		yield q[i % len(q)]
		q.rotate(-3 - i * 4)

def find_repeat(g):
	xs = []
	for i in count(1):
		xs.extend(islice(g, 10))
		if all((len(set(xs[j::i])) == 1 for j in range(i))):
			return ''.join(xs[:i])

def center_of(chars):
	return chars and find_repeat(iter_center(chars))


from re import sub
center_of = lambda c: sub('^(.+?)\\1*$', '\\g<1>', ''.join(((c + c)[(i * (i * 2 + 1) + i) % len(c)] for i in range(len(c)))))


def center_of(chars):
	size = len(chars)
	if size == 0:
		return ''
	size = len(chars)
	center_calc = size * 4
	center = ''
	index_large = 0
	for n in range(center_calc):
		index_large += n * 4
		index = index_large % size
		center += chars[index]
	for n in range(1, center_calc):
		if center[:n] == center[n:2 * n] and center[n:2 * n] == center[2 * n:3 * n] and (center[2 * n:3 * n] == center[3 * n:4 * n]):
			return center[:n]
	return center


def get_seq(string, slen):
	for i in range(1, slen + 1):
		sub = string[:i]
		times = slen // i
		if sub * times == string:
			return sub
	else:
		return False

def center_of(chars):
	if not chars:
		return chars
	seq = chars[0]
	chars_len = len(chars)
	i = 2
	lcount = 0
	while True:
		if i % 2:
			middle = chars[(lcount + i // 2) % chars_len]
			seq += middle
		else:
			lcount += 2 * i - 1
		i += 1
		lseq = len(seq)
		if lseq >= chars_len:
			subseq = get_seq(seq, lseq)
			if subseq:
				return subseq


def center_of(chars):
	center = ''
	n = len(chars)
	if n == 0:
		return center
	elif n % 2 == 0:
		n //= 2
	pos = []
	for i in range(n):
		pos.append(2 * i * (i + 1) % len(chars))
	for num in pos:
		center += chars[num]
	for i in range(len(center) // 2):
		test = center[:i + 1]
		if center == test * (len(center) // len(test)):
			center = test
			break
	return center


def center_of(chars):
	if chars == '':
		return ''
	chars = chars * 10000

	def n_int(n):
		return n * (n + 1) // 2
	center = ''
	c = 0
	i = 0
	while n_int(i) <= len(chars):
		if i % 2 != 0:
			c += 1
			pos_center = n_int(i) - c
			center += chars[pos_center]
		i += 1
	for x in range(1, len(center)):
		subcenter = center[:x]
		if subcenter * (len(center) // len(subcenter)) + subcenter[:len(center) % len(subcenter)] == center:
			return subcenter