Is case sensitivity important?

JavaScript (ES7), 43 bytes

f=(n,a=53,b=27)=>n?--n?f(n,a*63,b*37):a-b:0

<input type=number value=0 min=0 max=9 style=width:4ch oninput=o.textContent=f(+this.value)><div id=o>0</div>

Not reduces day today - 55 bytes.

TSQL, 41 bytes

DECLARE @ INT=10

PRINT 53*POWER(63.,@-1)-27*POWER(37.,@-1)

Fiddle

Jelly, 16 15 bytes

“~J“6j”OH*"’ÆḊḞ

Try it online! or verify all test cases.

How it works

“~J“6j”OH*"’ÆḊḞ  Main link. Argument: n

“~J“6j”          Yield ['~J', '6j'].
       O         Ordinal; replace the characters with it code points.
                 This yields [[126, 74], [54, 106]].
        H        Halve, yielding [[63, 37], [27, 53]].
           ’     Yield n-1.
         *"      Exponentiation (zipwith); elevate 63 and 37 to the power n-1.
            ÆḊ   Take the determinant of the matrix
                 [[63 ** (n-1), 37 ** (n-1)], [27, 53]], thus yielding
                 63 ** (n-1) * 53 - 37 ** (n-1) * 27.
              Ḟ  Floor; round the result down to the nearest integer.
                 This is required for n = 0.