Number of spanning trees in a wheel

A recurrence relation for the number of spanning trees
f(n)
in the wheel
W_{k},
where
n \geq 3
, is obtained as
f(n+1)-f(n)=L_{2^{n}+1},
where
f(3)=16
and where
L_{k}
is the
k
th number in the Lucas series
1, 3, 4, 7, \cdots , L_{k}, \cdots ,
where
L_{k} = L_{k+1} L_{k-1}
for
k > 1
. Alternately,
f(n) =L_{k}^{2} - 4 \delta
where
\delta = 0
for
n
odd and
1
for
n
even, thus confirming
f(n)
as a square number for
n
odd and serving to verify a previous finding in 1969 by Sedlacek that
f(n)=((3 + \sqrt{5})^{n} + (3 - \sqrt{5})^{n}/2^{n}-2
.