`0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5, 3, 0, 3, …`

This is the Van Eck’s Sequence.

Let’s go through it step by step.

Term 1: The first term is 0.

Term 2: Since we haven’t seen 0 before, the second term is 0.

Term 3: Since we had seen a 0 before, one step back, the third term is 1

Term 4: Since we haven’t seen a 1 before, the fourth term is 0

Term 5: Since we had seen a 0 before, two steps back, the fifth term is 2.

And so on…

Your task is to find the n_th number in Van Eck’s Sequence. (1-based)

## The Solution in Python

### Option 1

```
from collections import Counter
c=Counter()
SEQ = [0]
for i in range(1000):
n = SEQ[-1]
if not c[n]: c[n]=i
SEQ.append(i-c[n])
c[n]=i
seq=SEQ.__getitem__
```

### Option 2

```
def dist(arr):
for i in range (1, len(arr)):
if arr[-1-i] == arr[-1]:
return i
return 0
def seq(n):
s = [0, 0]
for _ in range (n):
s.append(dist(s))
return s[n-1]
```

```
def seq(n):
van, eck = [0], 0
while n := n - 1:
van.insert(0, eck := van.index(eck, 1) if eck in van[1:] else 0)
return eck
```

## Test cases to validate the solution

```
from solution import seq
import test
from random import randint
@test.describe("Sample tests:")
def tests():
@test.it("Small numbers")
def _():
s = [0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6]
for i in range (len(s)):
test.assert_equals(seq(i+1), s[i])
@test.it('Larger numbers')
def __():
s = [3, 1, 42, 0, 5, 15, 20, 0, 4, 32, 0, 3, 11,
18, 0, 4, 7, 0, 3, 7, 3, 2, 31, 0, 6, 31, 3,
6, 3, 2, 8, 33, 0, 9, 56, 0, 3, 8, 7, 19, 0,
5, 37, 0, 3, 8, 8, 1, 46, 0, 6, 23, 0]
for i in range (len(s)):
test.assert_equals(seq(i+50), s[i])
@test.describe('Random tests:')
def r():
def dist(arr):
for i in range (1, len(arr)):
if arr[-1-i] == arr[-1]:
return i
return 0
def ref_sol(n):
s = [0, 0]
for _ in range (n):
s.append(dist(s))
return s[n-1]
@test.it('200 random tests:')
def _():
for _ in range (200):
a = randint(100, 1000)
exp = ref_sol(a)
test.assert_equals(seq(a), exp)
```