Number spiral diagonals: https://projecteuler.net/problem=28

SumOfDiagonals.java

public class Solution {
  // The sum of each square is:
  //   `1`; n == 1
  //   `4n² + n²-(n-1) + n²-2(n-1) + n²-3(n-1)`; n > 1
  //     or `4n² - 6(n-1)`
  //
  //   Substituting `n = 2k+1`:
  //     `Σ(k ≤ K)(4(2k+1)² - 6(2k+1-1))`
  //     `Σ(k ≤ K)(4(4k²+4k+1) - 12k)`
  //     `4Σ(k ≤ K)(4k²+4k+1 - 3k)`
  //     `4Σ(k ≤ K)(4k² + k + 1)`
  //
  //  The total sum of the diagonals is therefore:
  //    `1 + 4Σ(1 ≤ k ≤ K)(4k² + k + 1)`; K = ½(N-1) and the first term is for N == 1
  //
  //  Substituting the closed form for the sum of sequence of squares, etc:
  //    `1 + 4(⅔K(K+1)(2K+1) + ½K(K+1) + K)`
  //    `1 + 4(⅙K(K+1)(8K+7) + K))`
  //
  //  Substituting back and simplifying:
  //    `1 + 4(⅙(½(N-1)(½(N-1)+1)(4(N-1)+7) + ½(N-1))`
  //    `1 + 4(½(N-1)½(N+1)⅙(4N+3) + ½(N-1))`
  //    `1 + 4(¼(N²-1)⅙(4N+3) + ½(N-1))`
  //    `1 + ⅙(N²-1)(4N+3) + 2(N-1)`
  //    `⅙(N²-1)(4N+3) + 2N - 1`
  public long sumOfDiagonals(long n) {
    return (n * n - 1) * (4 * n + 3) / 6 + 2 * n - 1;
  }
  
  @Test
  public void eyeballTest() {
    for (long n = 1; n < 1003; n += 2) {
      System.out.println("n = " + n + "; sOD(n) = " + sumOfDiagonals(n));
    }
  }
  
  public static void main(String[] args) {
    JUnitCore.main("Solution");
  }
}