BALKAN-SL 2015 N1

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$ , erases them and writes the number $1+ \sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$ .

(Albania)