A053854 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).

1, 9, 729, 531441, 3486784401, 205891132094649, 109418989131512359209, 523347633027360537213511521, 22528399544939174411840147874772641, 1394761471471951120984262893478242219427049, 601851824520496078935516587103606691779438596774649

Offset: 1

Author

Vladeta Jovovic, Mar 28 2000

Keywords

• nonn

Comments

Is this the same sequence (apart from the initial term) as A053764? - Philippe Deléham, Dec 09 2007

From M. F. Hasler, Oct 14 2008: (Start)

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.

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)

References

• V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A053774.

Extensions

More terms from Sean A. Irvine, Jan 16 2022