  A natural generalization of base B expansions is Zeckendorf's Theorem: every
integer can be uniquely written as a sum of non-consecutive Fibonacci numbers
$\{F_n\}$, with $F_{n+1} = F_n + F_{n-1}$ and $F_1=1, F_2=2$. If instead we
allow the coefficients of the Fibonacci numbers in the decomposition to be zero
or $\pm 1$, the resulting expression is known as the far-difference
representation. Alpert proved that a far-difference representation exists and
is unique under certain restraints that generalize non-consecutiveness,
specifically that two adjacent summands of the same sign must be at least 4
indices apart and those of opposite signs must be at least 3 indices apart. We
prove that a far-difference representation can be created using sets of
Skipponacci numbers, which are generated by recurrence relations of the form
$S^{(k)}_{n+1} = S^{(k)}_{n} + S^{(k)}_{n-k}$ for $k \ge 0$. Every integer can
be written uniquely as a sum of the $\pm S^{(k)}_n $'s such that every two
terms of the same sign differ in index by at least 2k+2, and every two terms of
opposite signs differ in index by at least k+2. Additionally, we prove that the
number of positive and negative terms in given Skipponacci decompositions
converges to a Gaussian, with a computable correlation coefficient that is a
rational function of the smallest root of the characteristic polynomial of the
recurrence. The proof uses recursion to obtain the generating function for
having a fixed number of summands, which we prove converges to the generating
function of a Gaussian. We next explore the distribution of gaps between
summands, and show that for any k the probability of finding a gap of length $j
\ge 2k+2$ decays geometrically, with decay ratio equal to the largest root of
the given k-Skipponacci recurrence. We conclude by finding sequences that have
an (s,d) far-difference representation for any positive integers s,d.
