A053846 Number of n X n matrices over GF(3) of order dividing 2 (i.e., number of solutions of X^2=I in GL(n,3)).

1, 2, 14, 236, 12692, 1783784, 811523288, 995733306992, 3988947598331024, 43581058503809001248, 1559669026899267564563936, 152805492791495918971070907584, 49094725258525117931062810300451648, 43237014297639482582550110281347475757696, 124920254287369111633119733942816364074145497472

Offset: 0

Author

Vladeta Jovovic, Mar 28 2000

Keywords

• nonn

Comments

Or, number of n X n invertible diagonalizable matrices over GF(3).

Examples

			a(2) = 14 because we have: {{0, 1}, {1, 0}}, {{0, 2}, {2, 0}}, {{1, 0}, {0, 1}}, {{1, 0}, {0,2}}, {{1, 0}, {1, 2}}, {{1, 0}, {2, 2}}, {{1, 1}, {0, 2}}, {{1,2}, {0, 2}}, {{2, 0}, {0, 1}}, {{2, 0}, {0, 2}}, {{2, 0}, {1,1}}, {{2, 0}, {2, 1}}, {{2, 1}, {0, 1}}, {{2, 2}, {0, 1}}. - _Geoffrey Critzer_, Aug 05 2017
		

References

• Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..60
• Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A053722, A053290, A290516.

Row sums of A378666.

Programs

Maple

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, T(n-1, k-1)+3^k*T(n-1, k)))
    end:
a:= n-> add(3^(k*(n-k))*T(n, k), k=0...n):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 06 2017

Mathematica

nn = 14; g[ n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
QFactorial[n, q]] /. q -> 3; G[z_] := Sum[z^k/g[k], {k, 0, nn}];Table[g[n], {n, 0, nn}] CoefficientList[Series[G[z]^2, {z, 0, nn}], z] (* Geoffrey Critzer, Aug 05 2017 *)

PARI

a(n)={my(v=[1]); for(n=1,n,v=vector(#v+1,k,if(k>1, v[k-1]) + if(k<=#v, 3^(k-1)*v[k]))); sum(k=0,n,3^(k*(n-k))*v[k+1])} \\ Andrew Howroyd, Mar 02 2018

Python

from sympy.core.cache import cacheit
@cacheit
def T(n, k): return 0 if k<0 or k>n else 1 if n==0 else T(n - 1, k - 1) + 3**k*T(n - 1, k)
def a(n): return sum(3**(k*(n - k))*T(n, k) for k in range(n + 1))
print([a(n) for n in range(15)]) # Indranil Ghosh, Aug 06 2017, after Maple code

Formula

a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^2. - Geoffrey Critzer, Aug 05 2017

Extensions

More terms from Geoffrey Critzer, Aug 05 2017