Question 64
The dragon Spas has one head. His family tree consists of Spas, the Spas parents, their parents, etc. It is known that if a dragon has $n$ heads, then his mother has $3 n$ heads and his father has $3 n+1$ heads. A positive integer is called good if it can be written in a unique way as a sum of the numbers of the heads of two dragons from the Spas' family tree. Prove that $2003$ is a good number and find the number of the good numbers less than $2003$.