Connell sequence in Python

The Connell sequence is a mathematical sequence that alternates between groups of consecutive odd and even integers, with each group size increasing by one. Understanding this pattern helps us find the nth term efficiently.

Understanding the Connell Sequence Pattern

The sequence follows this specific pattern ?

• Take first 1 odd integer: 1
• Take next 2 even integers: 2, 4
• Take next 3 odd integers: 5, 7, 9
• Take next 4 even integers: 10, 12, 14, 16
• Continue alternating with increasing group sizes

The complete sequence looks like: 1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, ...

Algorithm Steps

To find the nth term, we follow these steps ?

• Find which group contains the nth term
• Calculate the starting position of that group
• Determine if the group contains odd or even numbers
• Calculate the exact value at position n

Implementation

class Solution:
    def solve(self, n):
        i = 1
        # Find which group contains the nth term
        while (i * (i + 1) // 2) < n + 1:
            i += 1
        
        # Calculate group ending index and starting number
        idx = i * (i + 1) // 2
        num = i**2
        
        return num - 2 * (idx - n - 1)

# Test the solution
ob = Solution()
print(f"12th term: {ob.solve(12)}")
print(f"5th term: {ob.solve(5)}")
print(f"1st term: {ob.solve(1)}")

12th term: 21
5th term: 7
1st term: 1

How the Formula Works

The algorithm uses these key insights ?

• i * (i + 1) // 2 gives the cumulative count of elements up to group i
• i**2 represents the largest number in group i
• The final calculation num - 2 * (idx - n - 1) finds the exact term by working backwards from the group's end

Step-by-Step Example

For n = 12 (finding the 12th term) ?

def connell_detailed(n):
    print(f"Finding {n}th term:")
    
    i = 1
    while (i * (i + 1) // 2) < n + 1:
        cumulative = i * (i + 1) // 2
        print(f"Group {i} ends at position {cumulative}")
        i += 1
    
    print(f"Term {n} is in group {i}")
    
    idx = i * (i + 1) // 2
    num = i**2
    result = num - 2 * (idx - n - 1)
    
    print(f"Group {i} ends at position {idx}")
    print(f"Largest number in group {i}: {num}")
    print(f"Result: {result}")
    
    return result

connell_detailed(12)

Finding 12th term:
Group 1 ends at position 1
Group 2 ends at position 3
Group 3 ends at position 6
Group 4 ends at position 10
Term 12 is in group 5
Group 5 ends at position 15
Largest number in group 5: 25
Result: 21

Conclusion

The Connell sequence alternates between odd and even number groups with increasing sizes. The algorithm efficiently finds the nth term by determining which group contains it and calculating the position within that group using mathematical formulas.