RO-TST 1999 Problem 3

Prove that for any positive integer $n$ , the number $$S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n$$ is the sum of two consecutive perfect squares.

Dorin Andrica