The general form of the example given is

1 + 3 + 5 + \dots + (2 n - 1) + \dots + 5 + 3 + 1 = (n - 1)^{2} + n^{2}

, for $n = 1001$ : $1 + 3 + 5 + \dots + 1999 + 2001 + 1999 + \dots + 5 + 3 + 1 = 1000^{2} + 1001^{2} = 2002001$