Search for arithmetic progressions in rows of Pascal's triangle -- see http://math.stackexchange.com/questions/1083818.

Question1083818.java

import java.math.BigInteger;
import java.util.ArrayList;
import java.util.List;

public class Question1083818 {
	public static void main (String [] args) {
		List<BigInteger> pascal = new ArrayList<BigInteger> ();
		pascal.add (BigInteger.ONE);
		
		for (int n = 1;;n++) {
			List<BigInteger> next = new ArrayList<BigInteger> ();
			for (int k = 0;k <= n;k++)
				next.add ((k == 0 ? BigInteger.ZERO : pascal.get (k - 1)).add (k == n ? BigInteger.ZERO : pascal.get (k)));
			pascal = next;
			for (int i = 0;i + i <= n - 4;i++)
				for (int j = i + 1,k = i + 2;k + k <= n;k++) {
					BigInteger sum = pascal.get (i).add (pascal.get (k));
					if (!sum.testBit (0)) {
						sum = sum.shiftRight (1);
						int comparison; 
						while ((comparison = pascal.get (j).compareTo (sum)) < 0)
							j++;
						if (comparison == 0) {
							System.out.println (n + " : " + i + ", " + j + ", " + k + " : " + pascal.get (i) + ", " + pascal.get (j) + ", " + pascal.get (k));
							if (n != 19) {
								int a = (int) Math.sqrt (n + 4);
								int diff = a * a - n;
								boolean known = diff == 2 || diff == 4;
								if (known) {
									int a2 = (a * (a - 1)) / 2;
									known &= i == a2 - diff && j == a2 - diff / 2 && k == a2;
								}
								if (!known)
									throw new Error ();
							}
						}
					}
				}
		}
	}
}