Real Sequence/Examples/Arbitrary Sequence 1

Examples of Real Sequence
Consider the real sequence whose first few terms are:


 * $\dfrac 1 2, 1, -\dfrac 1 2, -1, \dfrac 1 4, \dfrac 1 2, \dotsc$

has no obvious formula to define it.

Discussion
The source work from which this example comes says:


 * In these examples, there is a simple formula for $s_n$ in terms of $n$, which the reader will spot.

Neither the author of this page nor his tutors for this part of his degree course were able to identify what that "simple" rule is.

Note that it is straightforward to design a rule which, while not being simple as such, is sufficient to do the job, for example:


 * $S = \sequence {\paren {-1} ^ {\ceiling {\frac n 2} - 1} \times 2 ^ {\ceiling {\frac n 4} - \paren {n - 1} \bmod 2} }$

or:
 * $s_n = \begin {cases} \dfrac 1 2 & : n = 1 \\ \dfrac {s_{n - 1} } {1 - n} & : \text {$n$ odd} \\ 2 s_{n - 1} & : \text {$n$ even}, n > 0 \end {cases}$