The function $f$ has the properties that $f(1)=6$ and $f(2 x+1)=3 f(x)$ for every integer $x$. What is the value of $f(63)$ ?
(A) 4374
(B) 1162
(C) 54
(D) 1458
(E) 486
We use the functional equation $f(2 x+1)=3 f(x)$ repeatedly.

Setting $x=1$, we get $f(3)=3 f(1)=3 \times 6=18$.

Setting $x=3$, we get $f(7)=3 f(3)=3 \times 18=54$.

Setting $x=7$, we get $f(15)=3 f(7)=3 \times 54=162$.

Setting $x=15$, we get $f(31)=3 f(15)=3 \times 162=486$.

Setting $x=31$, we get $f(63)=3 f(31)=3 \times 486=1458$.

ANSWER: (D)