Recovering highly-complex linear recurrences of integer sequences

Abstract We suggest a variant of the Berlekamp–Massey algorithm, originally used for recovering a linear shift register from known output bits, for recovering a linear recurrence satisfied by a sequence of natural numbers from known values of the sequence. We present an application of the algorithm to recovering extremely complex recurrences satisfied by the sequences enumerating polyominoes on twisted cylinders.

[1]  Heather J. Ruskin,et al.  Percolation processes in d-dimensions , 1976 .

[2]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[3]  N. J. A. Sloane,et al.  Shift-Register Synthesis (Modula m) , 1985, CRYPTO.

[4]  Andrei Asinowski,et al.  Formulae for Polyominoes on Twisted Cylinders , 2014, LATA.

[5]  Simon R. Blackburn,et al.  Fast rational interpolation, Reed-Solomon decoding, and the linear complexity profiles of sequences , 1997, IEEE Trans. Inf. Theory.

[6]  Günter Rote,et al.  Λ > 4: an Improved Lower Bound on the Growth Constant of Polyominoes , 2016, Commun. ACM.

[7]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[8]  James L. Massey,et al.  Shift-register synthesis and BCH decoding , 1969, IEEE Trans. Inf. Theory.

[9]  Neal Madras,et al.  A pattern theorem for lattice clusters , 1999 .

[10]  Iwan Jensen,et al.  Counting Polyominoes: A Parallel Implementation for Cluster Computing , 2003, International Conference on Computational Science.

[11]  D. Klarner.  Cell Growth Problems , 1967, Canadian Journal of Mathematics.

[12]  D. Klarner,et al.  A Procedure for Improving the Upper Bound for the Number of n-Ominoes , 1972, Canadian Journal of Mathematics - Journal Canadien de Mathematiques.

[13]  G. Rote,et al.  Counting Polyominoes on Twisted Cylinders , 2004 .

[14]  R. Blahut.  Theory and practice of error control codes , 1983 .