  Random matrix theory successfully models many systems, from the energy levels
of heavy nuclei to zeros of $L$-functions. While most ensembles studied have
continuous spectral distribution, Burkhardt et al introduced the ensemble of
$k$-checkerboard matrices, a variation of Wigner matrices with entries in
generalized checkerboard patterns fixed and constant. In this family, $N-k$ of
the eigenvalues are of size $O(\sqrt{N})$ and were called bulk while the rest
are tightly contrained around a multiple of $N$ and were called blip.
  We extend their work by allowing the fixed entries to take different constant
values. We can construct ensembles with blip eigenvalues at any multiples of
$N$ we want with any multiplicity (thus we can have the blips occur at
sequences such as the primes or the Fibonaccis). The presence of multiple blips
creates technical challenges to separate them and to look at only one blip at a
time. We overcome this by choosing a suitable weight function which allows us
to localize at each blip, and then exploiting cancellation to deal with the
resulting combinatorics to determine the average moments of the ensemble; we
then apply standard methods from probability to prove that almost surely the
limiting distributions of the matrices converge to the average behavior as the
matrix size tends to infinity. For blips with just one eigenvalue in the limit
we have convergence to a Dirac delta spike, while if there are $k$ eigenvalues
in a blip we again obtain hollow $k \times k$ GOE behavior.
