Geodesic
Formulas for the numbers of sequences containing a given pattern given number of times
Diskretnaya Matematika, Tome 32 (2020) no. 4, pp. 120-136

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Explicit recurrent formulas for the numbers of sequences containing a given pattern given number of times are constructed. These formulas depend on the length of the sequence, the length of the pattern and its period only. By means of these results one may find the distribution of statistics of the NIST overlapping matching test for binary sequences and arbitrary pattern parameters.
Keywords: statistical tests, binary sequence, pattern (sequence segment), recurrence relations. 
@article{DM_2020_32_4_a6,
     author = {A. A. Serov},
     title = {Formulas for the numbers of sequences containing a given pattern given number of times},
     journal = {Diskretnaya Matematika},
     pages = {120--136},
     publisher = {mathdoc},
     volume = {32},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2020_32_4_a6/}
}
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A. A. Serov. Formulas for the numbers of sequences containing a given pattern given number of times. Diskretnaya Matematika, Tome 32 (2020) no. 4, pp. 120-136. http://geodesic.mathdoc.fr/item/DM_2020_32_4_a6/