Recaman Sequence: Minimum Array Element Required to Change

Recaman sequence is a famous mathematical sequence that is defined as follows:

• a[0] = 0
• a[n] = a[n-1] - n if the result is positive and not already in the sequence; otherwise a[n] = a[n-1] + n.

For example, the first few terms of the Recaman sequence are as follows:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ...

In this challenge, we are given an array of integers and we need to find the minimum element that we can change so that the resulting array becomes the Recaman sequence.

Approach

To solve this challenge, we can generate the Recaman sequence up to the maximum value in the given array. Then, we can compare the generated sequence with the given array and find the first element that is not equal. This element will be the minimum element that we need to change to form the Recaman sequence.

Here is the Python code for this approach:

def recaman_sequence(n):
    a = [0]*n
    s = set()
    s.add(0)
    for i in range(1, n):
        c = a[i-1] - i
        if c < 0 or c in s:
            c = a[i-1] + i
        a[i] = c
        s.add(c)
    return a

def min_element_to_change(arr):
    max_val = max(arr)
    rec = recaman_sequence(max_val + 1) # generate Recaman sequence
    for i in range(len(arr)):
        if arr[i] != rec[i]:
            return arr[i]
    return -1 # not possible to change

# example usage
arr = [2, 7, 13, 20, 12, 21]
result = min_element_to_change(arr)
if result != -1:
    print(f"The minimum element to change is {result}.")
else:
    print("The given array cannot be converted to the Recaman sequence.")

Conclusion

In this challenge, we learned how to form the Recaman sequence and how to find the minimum element required to change a given array to the Recaman sequence. This approach can be easily extended to find the minimum number of changes required to form the Recaman sequence from a given array.