2579. Count Total Number of Colored Cells

Description

There exists an infinitely large two-dimensional grid of uncolored unit cells. You are given a positive integer n, indicating that you must do the following routine for n minutes:

• At the first minute, color any arbitrary unit cell blue.
• Every minute thereafter, color blue every uncolored cell that touches a blue cell.

Below is a pictorial representation of the state of the grid after minutes 1, 2, and 3.

Return the number of colored cells at the end of n minutes.

 

Example 1:

Input: n = 1
Output: 1
Explanation: After 1 minute, there is only 1 blue cell, so we return 1.

Example 2:

Input: n = 2
Output: 5
Explanation: After 2 minutes, there are 4 colored cells on the boundary and 1 in the center, so we return 5. 

 

Constraints:

• 1 <= n <= 105

 

Solutions

Solution: Math

• Time complexity: O(1)
• Space complexity: O(1)

 

JavaScript

/**
 * @param {number} n
 * @return {number}
 */
const coloredCells = function (n) {
  const lifting = 4;
  const base = 1;

  // Arithmetic progression
  // S = Number of terms / 2 × (first term + last term)

  return base + ((n - 1) / 2) * (lifting * n);
};