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%I #53 Sep 30 2023 18:30:07
%S 1,3,16,209,8739,1839102
%N Number of distinct characteristic polynomials of n X n matrices with elements {-1, 0, +1}.
%D Robert M. Corless, Bohemian Eigenvalues, Talk Presented at Computational Discovery in Mathematics (ACMES 2), University of Western Ontario, May 12 2016. (Talk based on joint work with Steven E. Thornton, Sonia Gupta, Jonathan Brino-Tarasoff, Venkat Balasubramanian.)
%H Eunice Y. S. Chan, <a href="https://ir.lib.uwo.ca/etd/6414">Algebraic Companions and Linearizations</a>, The University of Western Ontario (Canada, 2019) Electronic Thesis and Dissertation Repository. 6414.
%H Eunice Y. S. Chan and Robert Corless, <a href="https://doi.org/10.13001/1081-3810.3400">A new kind of companion matrix</a>, Electronic Journal of Linear Algebra, Volume 32, Article 25, 2017, see p. 335.
%H Robert M. Corless et al., <a href="https://web.archive.org/web/20230228075456/https://s3.amazonaws.com/stevenethornton.github/BHIME+Slides.pdf">Bohemian Eigenvalues</a>.
%H Robert Corless and Steven Thornton, <a href="http://www.bohemianmatrices.com/assets/images/posters/The_Bohemian_Eigenvalue_Project-FIRS_2017.pdf">The Bohemian Eigenvalue Project</a>, 2017 poster.
%F a(n) <= 3^(n^2). - _Robert P. P. McKone_, Sep 16 2023
%t a[n_] := a[n] = Module[{m, cPolys}, m = Tuples[Tuples[{-1, 0, 1}, n], n]; cPolys = CharacteristicPolynomial[#, x] & /@ m; Length[DeleteDuplicates[cPolys]]]; Table[a[i], {i, 1, 3}] (* _Robert P. P. McKone_, Sep 16 2023 *)
%o (Python)
%o from itertools import product
%o from sympy import Matrix
%o def A272658(n): return len({tuple(Matrix(n,n,p).charpoly().as_list()) for p in product((-1,0,1),repeat=n**2)}) if n else 1 # _Chai Wah Wu_, Sep 30 2023
%Y Six classes of matrices mentioned in Rob Corless's talk: this sequence, A272659, A272660, A272661, A272662, A272663.
%Y Other properties of this class of matrices: A271570, A271587, A271588. - _Steven E. Thornton_, Jul 13 2016
%K nonn,more,hard
%O 0,2
%A _N. J. A. Sloane_, May 15 2016
%E a(4) found by Daniel Lichtblau, May 13 2016
%E a(5) found by Daniel Lichtblau and _Steven E. Thornton_, May 19 2016
%E a(0)=1 prepended by _Alois P. Heinz_, Sep 28 2023
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