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%I #27 Jan 01 2024 14:19:15
%S 1,9,729,531441,3486784401,205891132094649,109418989131512359209,
%T 523347633027360537213511521,22528399544939174411840147874772641,
%U 1394761471471951120984262893478242219427049,601851824520496078935516587103606691779438596774649
%N Number of n X n matrices over GF(3) of order dividing 9, i.e., the number of solutions to X^9=I in GL(n,3).
%C Is this the same sequence (apart from the initial term) as A053764? - _Philippe Deléham_, Dec 09 2007
%C From _M. F. Hasler_, Oct 14 2008: (Start)
%C X^9 = I <=> I - X^9 = 0 <=> (I - X)^9 = 0 in GF(3). So to any solution of the first equation corresponds a solution X' = I-X of the other equation and vice versa. On the other hand, from considerations about the matrix rank (e.g., reasoning in Jordan basis) it is known that to check for nilpotency it is sufficient to go up to an exponent equal to the size of the matrix.
%C Thus by going out to the 9th power one finds all nilpotent matrices for sizes <= 9 X 9. Since A053854 is only given up to n=9, we can't see if A053764(10) is strictly bigger than A053854(10), which seems very likely since from then on there should be more matrices that satisfy A^10=0 than there are matrices satisfying A^9=0. (End)
%D V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
%H Kent E. Morrison, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%Y Cf. A053774.
%K nonn
%O 1,2
%A _Vladeta Jovovic_, Mar 28 2000
%E More terms from _Sean A. Irvine_, Jan 16 2022
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