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

The limiting spectral measure for an ensemble of generalized checkerboard matrices